IODS course project
Margot de Bazillac
Introduction to Open Data Science - Course Project
About the project
date()## [1] "Mon Dec 11 19:16:54 2023"Heads-up: I am new with R. During this first week, I
studied the 4 first chapters of the book made on R studio and watched
the given tutorial videos.
I have learned how to:
- set my directory
- use the R project mode, the R markdown file and github commit
- install and use libraries
- identify and use the different data types including the factors
- use the pipe %>% (it means “and then…”)
- open data set, read data set, read the structure of the data set
- select rows, columns and match text or rename column name
- filter data based on variable values
- create a table / tibble
- create an object that is reusable later
- use the time format
- add columns with text, int, calculations, categories etc.
- group data with one or many variables to then perform
calculation
- combine all data types to create a string with the function
paste()
- check the NA and remove the NA to do calculations
- summarize to sum up data per variable value
- manipulate table from wide to long and long to wide
- combine 2 tables (functions: bind, inner_join, full_join, left_join,
right_join)
- use the most common arithmetic functions and use percentage
- rank and reorder value
- create plots (bar, col, line, point, box, histogram)
- play with the different style and aesthetic possibilities
- add labels to plots
- combine plots together
With this tool I discovered how easy and fast it can be to manipulate
and visualize data from a big data set.
Conclusion of Week 1:
The book is really well made. The style is easy to follow and
understand. We can learn at our own pace and try the examples in R
studio to assimilate the concepts well.
I am new with R and I find it hard to learn so many new different
concepts in a week. I would have liked to have more exercises, without
the solutions directly written under and with a tool to correct our
lines of code. Finally, I understand it is not necessary to know by
heart every function, but I would like to understand well the
capabilities of this tool by practicing and doing more real life
exercises.
Lines of code to summarize learning in week 1:
# Get the data set
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.3 ✔ readr 2.1.4
## ✔ forcats 1.0.0 ✔ stringr 1.5.0
## ✔ ggplot2 3.4.4 ✔ tibble 3.2.1
## ✔ lubridate 1.9.3 ✔ tidyr 1.3.0
## ✔ purrr 1.0.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorslibrary(scales)##
## Attaching package: 'scales'
##
## The following object is masked from 'package:purrr':
##
## discard
##
## The following object is masked from 'package:readr':
##
## col_factorgbd_full <- read_csv("https://raw.githubusercontent.com/KimmoVehkalahti/RHDS/master/data/global_burden_disease_cause-year-sex-income.csv")## Rows: 168 Columns: 5
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (3): cause, sex, income
## dbl (2): year, deaths_millions
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# This is a code I wrote to know the pourcentage of deaths per cause for each year.
exercise4 <- gbd_full %>%
group_by(year, cause) %>%
summarise(total_per_cause= sum(deaths_millions)) %>%
group_by(year) %>%
mutate(total_per_year =sum(total_per_cause)) %>%
mutate(percentage = percent(total_per_cause/total_per_year)) %>%
select(year,cause,percentage) %>%
pivot_wider(names_from = cause, values_from = percentage)## `summarise()` has grouped output by 'year'. You can override using the
## `.groups` argument.exercise4## # A tibble: 7 × 4
## # Groups: year [7]
## year `Communicable diseases` Injuries `Non-communicable diseases`
## <dbl> <chr> <chr> <chr>
## 1 1990 33% 9% 58%
## 2 1995 31% 9% 60%
## 3 2000 29% 9% 62%
## 4 2005 27% 9% 64%
## 5 2010 24% 9% 67%
## 6 2015 20% 8% 72%
## 7 2017 19% 8% 73%# This is a code I wrote to know the number of deaths per sex and income and see the ratio Male / Female for each income type.
gbd_full %>%
filter(year ==1990) %>%
group_by(sex,income) %>%
summarize(deaths_per_sex_income = sum(deaths_millions)) %>%
pivot_wider(names_from = sex, values_from = deaths_per_sex_income) %>%
mutate(diff_M_F = Male/Female)## `summarise()` has grouped output by 'sex'. You can override using the `.groups`
## argument.## # A tibble: 4 × 4
## income Female Male diff_M_F
## <chr> <dbl> <dbl> <dbl>
## 1 High 4.14 4.46 1.08
## 2 Low 2.22 2.57 1.16
## 3 Lower-Middle 8.47 9.83 1.16
## 4 Upper-Middle 6.68 7.95 1.19Chapter 2: Regression and model validation
date()## [1] "Mon Dec 11 19:16:56 2023"# set directory
setwd("/Users/margot/Desktop/Desktop - MacBook Pro de MARGOT/Open data with R 2023/IODS-project")Thoughts about this week 2:
After reading all the chapters 1-7, I am now more confident to use R studio. I also understand the language better and I can do research on the web to use new function that I did not know.
It is very exciting to see how efficient is this tool and to think about all the analyzes we can do. I am an open university student and I can already see how to use this tool at work :).
Exercise: WRANGLING Please find the script file in the Github: create_learning2014_week2
Exercise: DATA ANALYSIS
QUESTION 1:
# Using the table from the course.
csv_table_read <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/learning2014.txt", sep = ",", header = T)
# library
library("tidyverse")
library("finalfit")
library("broom")
csv_table_read## gender age attitude deep stra surf points
## 1 F 53 3.7 3.583333 3.375 2.583333 25
## 2 M 55 3.1 2.916667 2.750 3.166667 12
## 3 F 49 2.5 3.500000 3.625 2.250000 24
## 4 M 53 3.5 3.500000 3.125 2.250000 10
## 5 M 49 3.7 3.666667 3.625 2.833333 22
## 6 F 38 3.8 4.750000 3.625 2.416667 21
## 7 M 50 3.5 3.833333 2.250 1.916667 21
## 8 F 37 2.9 3.250000 4.000 2.833333 31
## 9 M 37 3.8 4.333333 4.250 2.166667 24
## 10 F 42 2.1 4.000000 3.500 3.000000 26
## 11 M 37 3.9 3.583333 3.625 2.666667 31
## 12 F 34 3.8 3.833333 4.750 2.416667 31
## 13 F 34 2.4 4.250000 3.625 2.250000 23
## 14 F 34 3.0 3.333333 3.500 2.750000 25
## 15 M 35 2.6 4.166667 1.750 2.333333 21
## 16 F 33 4.1 3.666667 3.875 2.333333 31
## 17 F 32 2.6 4.083333 1.375 2.916667 20
## 18 F 44 2.6 3.500000 3.250 2.500000 22
## 19 M 29 1.7 4.083333 3.000 3.750000 9
## 20 F 30 2.7 4.000000 3.750 2.750000 24
## 21 M 27 3.9 3.916667 2.625 2.333333 28
## 22 M 29 3.4 4.000000 2.375 2.416667 30
## 23 F 31 2.7 4.000000 3.625 3.000000 24
## 24 F 37 2.3 3.666667 2.750 2.416667 9
## 25 F 26 3.7 3.666667 1.750 2.833333 26
## 26 F 26 4.4 4.416667 3.250 3.166667 32
## 27 M 30 4.1 3.916667 4.000 3.000000 32
## 28 F 33 3.7 3.750000 3.625 2.000000 33
## 29 F 33 2.5 3.250000 2.875 3.500000 29
## 30 M 28 3.0 3.583333 3.000 3.750000 30
## 31 M 26 3.4 4.916667 1.625 2.500000 19
## 32 F 27 3.2 3.583333 3.250 2.083333 23
## 33 F 25 2.0 2.916667 3.500 2.416667 19
## 34 F 31 2.4 3.666667 3.000 2.583333 12
## 35 M 20 4.2 4.500000 3.250 1.583333 10
## 36 F 39 1.6 4.083333 1.875 2.833333 11
## 37 M 38 3.1 3.833333 4.375 1.833333 20
## 38 M 24 3.8 3.250000 3.625 2.416667 26
## 39 M 26 3.8 2.333333 2.500 3.250000 31
## 40 M 25 3.3 3.333333 1.250 3.416667 20
## 41 F 30 1.7 4.083333 4.000 3.416667 23
## 42 F 25 2.5 2.916667 3.000 3.166667 12
## 43 M 30 3.2 3.333333 2.500 3.500000 24
## 44 F 48 3.5 3.833333 4.875 2.666667 17
## 45 F 24 3.2 3.666667 5.000 2.416667 29
## 46 F 40 4.2 4.666667 4.375 3.583333 23
## 47 M 25 3.1 3.750000 3.250 2.083333 28
## 48 F 23 3.9 3.416667 4.000 3.750000 31
## 49 F 25 1.9 4.166667 3.125 2.916667 23
## 50 F 23 2.1 2.916667 2.500 2.916667 25
## 51 M 27 2.5 4.166667 3.125 2.416667 18
## 52 M 25 3.2 3.583333 3.250 3.000000 19
## 53 M 23 3.2 2.833333 2.125 3.416667 22
## 54 F 23 2.6 4.000000 2.750 2.916667 25
## 55 F 23 2.3 2.916667 2.375 3.250000 21
## 56 F 45 3.8 3.000000 3.125 3.250000 9
## 57 F 22 2.8 4.083333 4.000 2.333333 28
## 58 F 23 3.3 2.916667 4.000 3.250000 25
## 59 M 21 4.8 3.500000 2.250 2.500000 29
## 60 M 21 4.0 4.333333 3.250 1.750000 33
## 61 F 21 4.0 4.250000 3.625 2.250000 33
## 62 F 21 4.7 3.416667 3.625 2.083333 25
## 63 F 26 2.3 3.083333 2.500 2.833333 18
## 64 F 25 3.1 4.583333 1.875 2.833333 22
## 65 F 26 2.7 3.416667 2.000 2.416667 17
## 66 M 21 4.1 3.416667 1.875 2.250000 25
## 67 F 23 3.4 3.416667 4.000 2.833333 28
## 68 F 22 2.5 3.583333 2.875 2.250000 22
## 69 F 22 2.1 1.583333 3.875 1.833333 26
## 70 F 22 1.4 3.333333 2.500 2.916667 11
## 71 F 23 1.9 4.333333 2.750 2.916667 29
## 72 M 22 3.7 4.416667 4.500 2.083333 22
## 73 M 23 3.2 4.833333 3.375 2.333333 21
## 74 M 24 2.8 3.083333 2.625 2.416667 28
## 75 F 22 4.1 3.000000 4.125 2.750000 33
## 76 F 23 2.5 4.083333 2.625 3.250000 16
## 77 M 22 2.8 4.083333 2.250 1.750000 31
## 78 M 20 3.8 3.750000 2.750 2.583333 22
## 79 M 22 3.1 3.083333 3.000 3.333333 31
## 80 M 21 3.5 4.750000 1.625 2.833333 23
## 81 F 22 3.6 4.250000 1.875 2.500000 26
## 82 F 23 2.6 4.166667 3.375 2.416667 12
## 83 M 21 4.4 4.416667 3.750 2.416667 26
## 84 M 22 4.5 3.833333 2.125 2.583333 31
## 85 M 29 3.2 3.333333 2.375 3.000000 19
## 86 F 29 3.9 3.166667 2.750 2.000000 30
## 87 F 21 2.5 3.166667 3.125 3.416667 12
## 88 M 28 3.3 3.833333 3.500 2.833333 17
## 89 F 21 3.3 4.250000 2.625 2.250000 18
## 90 F 30 3.0 3.833333 3.375 2.750000 19
## 91 F 21 2.9 3.666667 2.250 3.916667 21
## 92 M 23 3.3 3.833333 3.000 2.333333 24
## 93 F 21 3.3 3.833333 4.000 2.750000 28
## 94 F 21 3.5 3.833333 3.500 2.750000 17
## 95 F 20 3.6 3.666667 2.625 2.916667 18
## 96 M 22 3.7 4.333333 2.500 2.083333 17
## 97 M 21 4.2 3.750000 3.750 3.666667 23
## 98 M 21 3.2 4.166667 3.625 2.833333 26
## 99 F 20 5.0 4.000000 4.125 3.416667 28
## 100 M 22 4.7 4.000000 4.375 1.583333 31
## 101 F 20 3.6 4.583333 2.625 2.916667 27
## 102 F 20 3.6 3.666667 4.000 3.000000 25
## 103 M 24 2.9 3.666667 2.750 2.916667 23
## 104 F 20 3.5 3.833333 2.750 2.666667 21
## 105 F 19 4.0 2.583333 1.375 3.000000 27
## 106 F 21 3.5 3.500000 2.250 2.750000 28
## 107 F 21 3.2 3.083333 3.625 3.083333 23
## 108 F 22 2.6 4.250000 3.750 2.500000 21
## 109 F 25 2.0 3.166667 4.000 2.333333 25
## 110 F 21 2.7 3.083333 3.125 3.000000 11
## 111 F 22 3.2 4.166667 3.250 3.000000 19
## 112 F 25 3.3 2.250000 2.125 4.000000 24
## 113 F 20 3.9 3.333333 2.875 3.250000 28
## 114 M 24 3.3 3.083333 1.500 3.500000 21
## 115 F 20 3.0 2.750000 2.500 3.500000 24
## 116 M 21 3.7 3.250000 3.250 3.833333 24
## 117 F 20 2.5 4.000000 3.625 2.916667 20
## 118 F 20 2.9 3.583333 3.875 2.166667 19
## 119 M 31 3.9 4.083333 3.875 1.666667 30
## 120 F 20 3.6 4.250000 2.375 2.083333 22
## 121 F 22 2.9 3.416667 3.000 2.833333 16
## 122 F 22 2.1 3.083333 3.375 3.416667 16
## 123 M 21 3.1 3.500000 2.750 3.333333 19
## 124 M 22 4.0 3.666667 4.500 2.583333 30
## 125 F 21 3.1 4.250000 2.625 2.833333 23
## 126 F 21 2.3 4.250000 2.750 3.333333 19
## 127 F 21 2.8 3.833333 3.250 3.000000 18
## 128 F 21 3.7 4.416667 4.125 2.583333 28
## 129 F 20 2.6 3.500000 3.375 2.416667 21
## 130 F 21 2.4 3.583333 2.750 3.583333 19
## 131 F 25 3.0 3.666667 4.125 2.083333 27
## 132 M 21 2.8 2.083333 3.250 4.333333 24
## 133 F 24 2.9 4.250000 2.875 2.666667 21
## 134 F 20 2.4 3.583333 2.875 3.000000 20
## 135 M 21 3.1 4.000000 2.375 2.666667 28
## 136 F 20 1.9 3.333333 3.875 2.166667 12
## 137 F 20 2.0 3.500000 2.125 2.666667 21
## 138 F 18 3.8 3.166667 4.000 2.250000 28
## 139 F 21 3.4 3.583333 3.250 2.666667 31
## 140 F 19 3.7 3.416667 2.625 3.333333 18
## 141 F 21 2.9 4.250000 2.750 3.500000 25
## 142 F 20 2.3 3.250000 4.000 2.750000 19
## 143 M 21 4.1 4.416667 3.000 2.000000 21
## 144 F 20 2.7 3.250000 3.375 2.833333 16
## 145 F 21 3.5 3.916667 3.875 3.500000 7
## 146 F 20 3.4 3.583333 3.250 2.500000 21
## 147 F 18 3.2 4.500000 3.375 3.166667 17
## 148 M 22 3.3 3.583333 4.125 3.083333 22
## 149 F 22 3.3 3.666667 3.500 2.916667 18
## 150 M 24 3.5 2.583333 2.000 3.166667 25
## 151 F 19 3.2 4.166667 3.625 2.500000 24
## 152 F 20 3.1 3.250000 3.375 3.833333 23
## 153 F 20 2.8 4.333333 2.125 2.250000 23
## 154 F 17 1.7 3.916667 4.625 3.416667 26
## 155 M 19 1.9 2.666667 2.500 3.750000 12
## 156 F 20 3.5 3.083333 2.875 3.000000 32
## 157 F 20 2.4 3.750000 2.750 2.583333 22
## 158 F 20 2.1 4.166667 4.000 3.333333 20
## 159 F 20 2.9 4.166667 2.375 2.833333 21
## 160 F 19 1.9 3.250000 3.875 3.000000 23
## 161 F 19 2.0 4.083333 3.375 2.833333 20
## 162 F 22 4.2 2.916667 1.750 3.166667 28
## 163 M 35 4.1 3.833333 3.000 2.750000 31
## 164 F 18 3.7 3.166667 2.625 3.416667 18
## 165 F 19 3.6 3.416667 2.625 3.000000 30
## 166 M 21 1.8 4.083333 3.375 2.666667 19# analyze the structure of the dataset
str(csv_table_read)## 'data.frame': 166 obs. of 7 variables:
## $ gender : chr "F" "M" "F" "M" ...
## $ age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: num 3.7 3.1 2.5 3.5 3.7 3.8 3.5 2.9 3.8 2.1 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ points : int 25 12 24 10 22 21 21 31 24 26 ...# analyze the dimension of the dataset
dim(csv_table_read)## [1] 166 7# Missing data? No data is missing.
ff_glimpse(csv_table_read)## $Continuous
## label var_type n missing_n missing_percent mean sd min
## age age <int> 166 0 0.0 25.5 7.8 17.0
## attitude attitude <dbl> 166 0 0.0 3.1 0.7 1.4
## deep deep <dbl> 166 0 0.0 3.7 0.6 1.6
## stra stra <dbl> 166 0 0.0 3.1 0.8 1.2
## surf surf <dbl> 166 0 0.0 2.8 0.5 1.6
## points points <int> 166 0 0.0 22.7 5.9 7.0
## quartile_25 median quartile_75 max
## age 21.0 22.0 27.0 55.0
## attitude 2.6 3.2 3.7 5.0
## deep 3.3 3.7 4.1 4.9
## stra 2.6 3.2 3.6 5.0
## surf 2.4 2.8 3.2 4.3
## points 19.0 23.0 27.8 33.0
##
## $Categorical
## label var_type n missing_n missing_percent levels_n levels
## gender gender <chr> 166 0 0.0 2 -
## levels_count levels_percent
## gender - -# summary statistics for each variable
missing_glimpse(csv_table_read)## label var_type n missing_n missing_percent
## gender gender <chr> 166 0 0.0
## age age <int> 166 0 0.0
## attitude attitude <dbl> 166 0 0.0
## deep deep <dbl> 166 0 0.0
## stra stra <dbl> 166 0 0.0
## surf surf <dbl> 166 0 0.0
## points points <int> 166 0 0.0# Count per gender and percentage male / female
library("scales")
csv_table_read %>%
count(gender) %>%
mutate(total_percentage = n / nrow(csv_table_read)) %>%
mutate(total_percentage2 = percent(total_percentage))## gender n total_percentage total_percentage2
## 1 F 110 0.6626506 66%
## 2 M 56 0.3373494 34%# Mean and median for exercises points, and learning method per gender
summary(csv_table_read)## gender age attitude deep
## Length:166 Min. :17.00 Min. :1.400 Min. :1.583
## Class :character 1st Qu.:21.00 1st Qu.:2.600 1st Qu.:3.333
## Mode :character Median :22.00 Median :3.200 Median :3.667
## Mean :25.51 Mean :3.143 Mean :3.680
## 3rd Qu.:27.00 3rd Qu.:3.700 3rd Qu.:4.083
## Max. :55.00 Max. :5.000 Max. :4.917
## stra surf points
## Min. :1.250 Min. :1.583 Min. : 7.00
## 1st Qu.:2.625 1st Qu.:2.417 1st Qu.:19.00
## Median :3.188 Median :2.833 Median :23.00
## Mean :3.121 Mean :2.787 Mean :22.72
## 3rd Qu.:3.625 3rd Qu.:3.167 3rd Qu.:27.75
## Max. :5.000 Max. :4.333 Max. :33.00# The age varies from 17 to 55, mean is 25 and median 22. it suggests that there are some relatively higher values in the dataset
# The attitude varies from 1.4 to 5
# The points are from 7 to 33 and the mean is 22 and the median is 23. It suggests that there are some relatively lower values in the dataset
# we analyze the variables for both genders and females
# draw a scatter plot matrix of the variables in learning2014.
# [-1] excludes the first column (gender)
pairs(csv_table_read[-1])
# access the GGally and ggplot2 libraries
library(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(ggplot2)
# create a more advanced plot matrix with ggpairs()
p <- ggpairs(csv_table_read, mapping = aes(), lower = list(combo = wrap("facethist", bins = 20)))
p
# some data shows that there could be correlation between some variables
# Relationship between points and attitudes
csv_table_read %>%
ggplot(aes(x= attitude, y= points)) +
geom_point() +
facet_wrap(~ gender) +
geom_smooth(method = "lm")## `geom_smooth()` using formula = 'y ~ x'
QUESTION 2, 3 and 4:
Female model
#female model
female_data <- csv_table_read %>%
filter(gender == "F")
View(female_data)
# Fit a multiple linear model for females. Let's check how points are influenced by age, attitude and deep learning approach
female_fitmodel <- lm(points ~ age + attitude + deep, data = female_data)
# In this model I want to check if age, attitude and deep impact points without impacting each other.
# summary of std, p value and
summary(female_fitmodel) ##
## Call:
## lm(formula = points ~ age + attitude + deep, data = female_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.058 -3.263 0.622 4.003 10.533
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.59029 4.28982 3.168 0.00201 **
## age -0.01743 0.06983 -0.250 0.80338
## attitude 3.40151 0.70837 4.802 5.19e-06 ***
## deep -0.27355 0.96270 -0.284 0.77685
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.353 on 106 degrees of freedom
## Multiple R-squared: 0.1791, Adjusted R-squared: 0.1559
## F-statistic: 7.71 on 3 and 106 DF, p-value: 0.0001043summary(female_fitmodel)$r.squared## [1] 0.1791183#"age," "attitude," and "deep" explains about 18% of the variation of "points"
# p value intercept: it is significant as very small (0.002) and seems to play a significant role in the regression model
# baseline of model in 13.59 (estimate), when no factors are taken into account.
# age is not significant and is not correlated with points
# deep is not significant and is not correlated with points
# attitude is significant and it seems to play a significant role on the points.
# for one point increase in the attitude, the points increase by 3.63 (estimate)
# High std shows that the estimate is not so precise. It could due to sample size.
# I decide to drop the deep and the age variables and keep only the attitude.
female_fitmodel2 <- lm(points ~ attitude, data = female_data)
summary(female_fitmodel2) ##
## Call:
## lm(formula = points ~ attitude, data = female_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.0557 -3.3486 0.6137 3.9819 10.3668
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.194 2.156 5.655 1.29e-07 ***
## attitude 3.389 0.701 4.835 4.44e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.307 on 108 degrees of freedom
## Multiple R-squared: 0.1779, Adjusted R-squared: 0.1703
## F-statistic: 23.38 on 1 and 108 DF, p-value: 4.442e-06tidy(female_fitmodel2)## # A tibble: 2 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 12.2 2.16 5.66 0.000000129
## 2 attitude 3.39 0.701 4.83 0.00000444summary(female_fitmodel2)$r.squared## [1] 0.1779266# p value is very low, same for the std, so this model is correct and justify the positive relation vs a positive attitude -> more points.
# rsquare is still quite low..
# The model doesn't provide a good fit for the data, and a significant portion of the variance is not explained. Is could be due to the sample size.
# autoplot: Residuals vs Fitted values, Normal QQ-plot and Residuals vs Leverage
# Identify issues with my regression model, such as non-linearity, non-normality, or influential data points
# autoplot doesnt knit.
#autoplot(female_fitmodel)
#autoplot(female_fitmodel2)
# I use plot function and which to get the desired plots.
plot(female_fitmodel,which = c(1,2,5))
plot(female_fitmodel2,which = c(1,2,5))
# we observe non normality at the end and beginning of the line in qq plot
# both models show that there are some points that are high leverage indicated on the residuals vs leverageMale model
# male model
male_data <- csv_table_read %>%
filter(gender == "M")
View(male_data)
summary(male_data)## gender age attitude deep
## Length:56 Min. :19.0 Min. :1.700 Min. :2.083
## Class :character 1st Qu.:21.0 1st Qu.:3.100 1st Qu.:3.396
## Mode :character Median :24.0 Median :3.400 Median :3.792
## Mean :26.8 Mean :3.443 Mean :3.725
## 3rd Qu.:29.0 3rd Qu.:3.900 3rd Qu.:4.083
## Max. :55.0 Max. :4.800 Max. :4.917
## stra surf points
## Min. :1.250 Min. :1.583 Min. : 9.00
## 1st Qu.:2.375 1st Qu.:2.312 1st Qu.:20.00
## Median :3.000 Median :2.625 Median :23.50
## Mean :2.964 Mean :2.704 Mean :23.48
## 3rd Qu.:3.531 3rd Qu.:3.167 3rd Qu.:28.25
## Max. :4.500 Max. :4.333 Max. :33.00# Fit a multiple linear model for males. Let's check how points are influenced by age, attitude and deep learning approach
male_fitmodel <- lm(points ~ age + attitude + deep, data = male_data)
# summary of std, p value and
summary(male_fitmodel) ##
## Call:
## lm(formula = points ~ age + attitude + deep, data = male_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.8084 -3.3162 -0.0696 3.2195 9.9927
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.21897 6.06857 3.002 0.004112 **
## age -0.16602 0.08456 -1.963 0.054974 .
## attitude 4.31829 1.11699 3.866 0.000309 ***
## deep -1.38378 1.22006 -1.134 0.261916
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.271 on 52 degrees of freedom
## Multiple R-squared: 0.2718, Adjusted R-squared: 0.2298
## F-statistic: 6.47 on 3 and 52 DF, p-value: 0.000835tidy(male_fitmodel)## # A tibble: 4 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 18.2 6.07 3.00 0.00411
## 2 age -0.166 0.0846 -1.96 0.0550
## 3 attitude 4.32 1.12 3.87 0.000309
## 4 deep -1.38 1.22 -1.13 0.262summary(male_fitmodel)$r.squared ## [1] 0.2718164# similar results than for the female.
# All variables have a smaller p value than for in the female model.
# rsquare is higher as it explains 27% but it is still quite low. It could be due to the sample size.
# I decide to drop the deep and the age as variables and keep only the attitude.
male_fitmodel2 <- lm(points ~ attitude, data = male_data)
summary(male_fitmodel2) ##
## Call:
## lm(formula = points ~ attitude, data = male_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.6535 -2.9073 -0.5121 3.6974 10.2106
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.061 3.953 2.292 0.02581 *
## attitude 4.189 1.129 3.711 0.00049 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.411 on 54 degrees of freedom
## Multiple R-squared: 0.2032, Adjusted R-squared: 0.1884
## F-statistic: 13.77 on 1 and 54 DF, p-value: 0.0004897tidy(male_fitmodel2)## # A tibble: 2 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 9.06 3.95 2.29 0.0258
## 2 attitude 4.19 1.13 3.71 0.000490summary(male_fitmodel)$r.squared ## [1] 0.2718164# p value is very low, same for the std, so this model is correct and justify the positive relation vs a positive attitude -> more points.
# rsquare is higher as it explains 27% but it is still quite low
# The model doesn't provide a good fit for the data, and a significant portion of the variance is not explained. Is could be due to the sample size.
# autoplot: Residuals vs Fitted values, Normal QQ-plot and Residuals vs Leverage
# Identify issues with my regression model, such as non-linearity, non-normality, or influential data points
# autoplot doesnt knit !!
# autoplot(male_fitmodel)
# autoplot(male_fitmodel2)
# plot with the plot function
plot(male_fitmodel,which = c(1,2,5))
plot(male_fitmodel2,which = c(1,2,5))
#The red line in residuals vs fitted stays quite close to the 0 line which is good
# both models show non normality. it is observed at the beginning of the qq plot
# both models show that there are some points that are high leverage indicated on the residuals vs leverageother models tested:
test_fit1 <- csv_table_read %>%
lm(points ~ deep, data = .)
library(ggfortify)
summary(test_fit1)##
## Call:
## lm(formula = points ~ deep, data = .)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.6913 -3.6935 0.2862 4.9957 10.3537
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 23.1141 3.0908 7.478 4.31e-12 ***
## deep -0.1080 0.8306 -0.130 0.897
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.913 on 164 degrees of freedom
## Multiple R-squared: 0.000103, Adjusted R-squared: -0.005994
## F-statistic: 0.01689 on 1 and 164 DF, p-value: 0.8967tidy(test_fit1) # p value is small and significant## # A tibble: 2 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 23.1 3.09 7.48 4.31e-12
## 2 deep -0.108 0.831 -0.130 8.97e- 1summary(test_fit1)$r.squared # too low## [1] 0.0001029919test_fit2 <- csv_table_read %>%
lm(points ~ deep * gender, data = .)
library(ggfortify)
summary(test_fit2)##
## Call:
## lm(formula = points ~ deep * gender, data = .)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.3247 -3.3338 0.3369 4.6242 10.6787
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 22.36387 3.91828 5.708 5.33e-08 ***
## deep -0.01001 1.06032 -0.009 0.992
## genderM 2.67476 6.41719 0.417 0.677
## deep:genderM -0.40787 1.71487 -0.238 0.812
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.922 on 162 degrees of freedom
## Multiple R-squared: 0.00922, Adjusted R-squared: -0.009127
## F-statistic: 0.5025 on 3 and 162 DF, p-value: 0.6811tidy(test_fit2) # p value is small and significant## # A tibble: 4 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 22.4 3.92 5.71 0.0000000533
## 2 deep -0.0100 1.06 -0.00944 0.992
## 3 genderM 2.67 6.42 0.417 0.677
## 4 deep:genderM -0.408 1.71 -0.238 0.812summary(test_fit2)$r.squared # too low## [1] 0.009220341Basic data
# Female vs Male participants
csv_table_read %>%
ggplot(aes(x=gender)) +
geom_bar()
# age chart and gender per age
csv_table_read %>%
ggplot(aes(x= age, fill = gender)) +
geom_bar()
# age chart distribution per gender
csv_table_read %>%
ggplot(aes(x= age)) +
facet_grid(~gender) +
geom_histogram()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# age box plot distribution per gender
csv_table_read %>%
ggplot(aes(x= gender, y=age)) +
geom_boxplot()
# relationship and distribution between age, points, and gender
csv_table_read %>%
ggplot(aes(y = points, x = age, colour = gender)) +
geom_point() +
labs(title = "Distribution of points per age and gender")
# with this data we can observe the different age points that drives the mean up (vs the median).Gender analysis
Gender and points
# Distribution of the points per gender - histogram
csv_table_read %>%
ggplot(aes(x = points)) +
geom_histogram() +
facet_grid(~gender) +
labs(title = "Histogram of points by Gender")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
#Distribution of the points per gender - boxplot
csv_table_read %>%
ggplot(aes(y = points, x = gender, colour = gender)) +
geom_boxplot() +
labs(title = "Boxplot of points by Gender")
#QQ plot - points per gender
csv_table_read %>%
ggplot(aes(sample = points)) +
geom_qq() +
geom_qq_line(colour = "blue") +
facet_grid(~gender)
# mean points per gender - this is not significant
csv_table_read %>%
t.test(points ~ gender, data = .)##
## Welch Two Sample t-test
##
## data: points by gender
## t = -1.1832, df = 107.84, p-value = 0.2393
## alternative hypothesis: true difference in means between group F and group M is not equal to 0
## 95 percent confidence interval:
## -3.0896905 0.7799502
## sample estimates:
## mean in group F mean in group M
## 22.32727 23.48214Gender and attitude
# attitude vs gender
csv_table_read %>%
ggplot(aes(x=gender, y= attitude)) +
geom_boxplot()
# Type histogram
csv_table_read %>%
ggplot(aes(x = attitude)) +
geom_histogram() +
facet_grid(~ gender) +
labs(title = "Histogram of attitude by Gender")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# QQ plot: attitude per gender
csv_table_read %>%
ggplot(aes(sample = attitude)) +
geom_qq() +
geom_qq_line(colour = "blue") +
facet_grid(~gender)
# mean attitude per gender - This is significant and shows a difference between F and M on deep
csv_table_read %>%
t.test(attitude ~ gender, data = .)##
## Welch Two Sample t-test
##
## data: attitude by gender
## t = -4.0932, df = 122.66, p-value = 7.657e-05
## alternative hypothesis: true difference in means between group F and group M is not equal to 0
## 95 percent confidence interval:
## -0.6718635 -0.2338508
## sample estimates:
## mean in group F mean in group M
## 2.990000 3.442857Gender and deep learning approach:
# deep learning approach vs gender
# We could do that for all approach of learning
# Type histogram
csv_table_read %>%
ggplot(aes(x = deep)) +
geom_histogram() +
facet_grid(~ gender) +
labs(title = "Histogram of deep approach by Gender")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
#Type boxplot
csv_table_read %>%
ggplot(aes(y = deep, x = gender, fill = gender)) +
geom_boxplot() +
labs(title = "Boxplot of deep Approach by Gender")
# QQ plot: deep per gender
csv_table_read %>%
ggplot(aes(sample = deep)) +
geom_qq() +
geom_qq_line(colour = "blue") +
facet_grid(~gender)
# mean deep per gender - This is quite significant and could show a correlation between the gender and the approach deep
csv_table_read %>%
t.test(deep ~ gender, data = .)##
## Welch Two Sample t-test
##
## data: deep by gender
## t = -0.72082, df = 101.32, p-value = 0.4727
## alternative hypothesis: true difference in means between group F and group M is not equal to 0
## 95 percent confidence interval:
## -0.2546963 0.1189279
## sample estimates:
## mean in group F mean in group M
## 3.656818 3.724702Age Analysis
Age and points
# does not seem to impact on points
csv_table_read %>%
ggplot(aes(x= age, y=points)) +
geom_point()+
facet_wrap(~ gender) +
geom_smooth(method = "lm")## `geom_smooth()` using formula = 'y ~ x'
Age and attitude
# does not seem to impact on attitude
csv_table_read %>%
ggplot(aes(x= age, y=attitude)) +
geom_point() +
facet_wrap(~ gender) +
geom_smooth(method = "lm")## `geom_smooth()` using formula = 'y ~ x'
Learning approach: deep analysis
deep and age
# deep learning approach vs age - no correlation
# We could do that for all approach of learning
csv_table_read %>%
ggplot(aes(x= age, y= deep)) +
geom_point() +
facet_wrap(~ gender) +
geom_smooth(method = "lm")## `geom_smooth()` using formula = 'y ~ x'
deep and points
# deep learning approach vs points - The deep approach seems to have a correlation with the number of points
csv_table_read %>%
ggplot(aes(x= deep, y=points)) +
geom_point() +
facet_wrap(~ gender) +
geom_smooth(method = "lm")## `geom_smooth()` using formula = 'y ~ x'
test_fit <- csv_table_read %>%
lm(points ~ deep * gender, data = .)
library(ggfortify)
summary(test_fit)##
## Call:
## lm(formula = points ~ deep * gender, data = .)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.3247 -3.3338 0.3369 4.6242 10.6787
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 22.36387 3.91828 5.708 5.33e-08 ***
## deep -0.01001 1.06032 -0.009 0.992
## genderM 2.67476 6.41719 0.417 0.677
## deep:genderM -0.40787 1.71487 -0.238 0.812
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.922 on 162 degrees of freedom
## Multiple R-squared: 0.00922, Adjusted R-squared: -0.009127
## F-statistic: 0.5025 on 3 and 162 DF, p-value: 0.6811# deep seems to have a significant impact on "points."Chapter 3
setwd("/Users/margot/Desktop/Desktop - MacBook Pro de MARGOT/Open data with R 2023/IODS-project")
library(broom)
library(ggplot2)
library(dplyr)
data <- read.csv("alc.csv",sep = ",", header = T)
class(data$high_use)## [1] "logical"class(data$famrel)## [1] "integer"class(data$goout)## [1] "integer"class(data$studytime)## [1] "integer"Question 3, 4 and 5
The purpose of your analysis is to study the relationships between high/low alcohol consumption and some of the other variables in the data. To do this, choose 4 interesting variables in the data and for each of them, present your personal hypothesis about their relationships with alcohol consumption. (0-1 point)
Hypothesis 1: students with a good family relationship have less chance to have a high consumption of alcohol
Comments on the below codes chunk: There is a negative correlation between family relationship and the consumption of alcohol. When the relationship is not good (from 1 to 5), there are more risk to drink lot of alcohol. Pvalue is 0,0204 which is representative(<0.05). The coefficient for the variable Famrel in this model is -0.2838. The odds ratio associated with a one-unit change in Famrel is approximately exp(-0.2838) = 0.753. For a one-unit decrease in Famrel, the odds of ‘high_use’ decrease by about 1 - 0.753 = 24.7%. With 95% confidence, the true value of the odds ratio for the intercept lies between approximately 0.49 and 3.35. For the famrel, the odds ratio lies between 0.59 and 0.95, since it is less than 1, there is a statistical significance, with a negative effect from the variable famrel to the high_use. The null deviance is bigger than the residual deviance of the model. It shows that the model explains a part of the variable high-use.
# I add the family = binomial as we do the analyze on a binary variable (True and False)
logistic_model_family_alcohol <- glm(high_use ~ famrel, data = data, family = binomial)
summary(logistic_model_family_alcohol)##
## Call:
## glm(formula = high_use ~ famrel, family = binomial, data = data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.2573 0.4853 0.530 0.5960
## famrel -0.2838 0.1224 -2.318 0.0204 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.04 on 369 degrees of freedom
## Residual deviance: 446.67 on 368 degrees of freedom
## AIC: 450.67
##
## Number of Fisher Scoring iterations: 4# get the coef
coef(logistic_model_family_alcohol) %>% exp()## (Intercept) famrel
## 1.2934265 0.7528865# odd of being in high use - exponential of the coef
exp(coef(logistic_model_family_alcohol))## (Intercept) famrel
## 1.2934265 0.7528865# get the intervals
confint(logistic_model_family_alcohol) %>% exp()## Waiting for profiling to be done...## 2.5 % 97.5 %
## (Intercept) 0.4961949 3.3547129
## famrel 0.5911258 0.9570208# get the variance, BIC and AIC
logistic_model_family_alcohol %>%
glance()## # A tibble: 1 × 8
## null.deviance df.null logLik AIC BIC deviance df.residual nobs
## <dbl> <int> <dbl> <dbl> <dbl> <dbl> <int> <int>
## 1 452. 369 -223. 451. 458. 447. 368 370# If I want to check the impact of high use to family relationship -> famrel is ordinal so I use the ordinal regression model
library(ordinal)##
## Attaching package: 'ordinal'## The following object is masked from 'package:dplyr':
##
## sliceordinal_model_family_alcohol <- clm(factor(famrel) ~ high_use, data = data)
summary(ordinal_model_family_alcohol)## formula: factor(famrel) ~ high_use
## data: data
##
## link threshold nobs logLik AIC niter max.grad cond.H
## logit flexible 370 -454.70 919.40 5(0) 3.77e-08 2.0e+01
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## high_useTRUE -0.5411 0.2139 -2.529 0.0114 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Threshold coefficients:
## Estimate Std. Error z value
## 1|2 -4.0079 0.3672 -10.914
## 2|3 -2.7746 0.2194 -12.645
## 3|4 -1.3118 0.1422 -9.222
## 4|5 0.8468 0.1300 6.516# This box plot confirms our hypothesis and conclusion. We can see that the mean and median of the family relationship quality is very different whether the students drink a lot or not.
data %>%
ggplot(aes(x= factor(high_use), y= famrel)) +
geom_boxplot() +
labs(title = "Quality of family relationship vs the alcohol consumption",
x = "High consumption",
y = "quality of family relationship")
data %>%
ggplot(aes(x = as.factor(famrel), fill = as.factor(high_use))) +
geom_bar(position = "fill") +
labs(
title = "Relationship Between Family relationship and Alcohol Consumption",
x = "Family relation score from 1 to 5",
y = "Proportion of students",
fill = "High Use of alcohol: True of False"
)
# there are less score 5 and way more score 2 among the students who drink a lot
data %>%
ggplot(aes(x = as.factor(high_use), fill = as.factor(famrel))) +
geom_bar(position = "fill") +
labs(
title = "Family relation score distribution by Alcohol Consumption",
x = "High Use of alcohol: True of False",
y = "Proportion of students",
fill = "Family relation score from 1 to 5"
)
Hypothesis 2: students who go more out with their friends have more risk to have a high consumption of alcohol
Comments on the below codes chunk: There is a a positive relationship between going out and the consumption of alcohol and a significant p value (very small value), which confirms the hypothesis. The more students go out the more risks they have to have a high consumption in alcohol.
The coefficient for goout is 0.7563.The odds ratio associated with a one-unit change in Goout is approximately exp(0.7563) = 2.13.For a one-unit increase in Goout, the odds of ‘high_use’ increase by about 2.13 - 1 = 1.13, which is about 113% increase.
With 95% confidence, the true value of the odds ratio for the intercept lies between approximately 0.015 and 0.078. For the goout, the odds ratio lies between 1.71 and 2.66, since it is > 1, there is a statistical significance with a positive effect of this variable on the high_use.
The null deviance is way bigger than the residual deviance of the model. It shows that the model explains the variable high-use, even better than in the hypothesis 1. The AIC of this model is even smaller than the previous model too, which confirms my conclusion.
logistic_model_goout_alcohol <- glm(high_use ~ goout, data = data, family = binomial)
summary(logistic_model_goout_alcohol)##
## Call:
## glm(formula = high_use ~ goout, family = binomial, data = data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -3.3368 0.4188 -7.968 1.62e-15 ***
## goout 0.7563 0.1165 6.494 8.34e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.04 on 369 degrees of freedom
## Residual deviance: 403.05 on 368 degrees of freedom
## AIC: 407.05
##
## Number of Fisher Scoring iterations: 4tidy(logistic_model_goout_alcohol)## # A tibble: 2 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) -3.34 0.419 -7.97 1.62e-15
## 2 goout 0.756 0.116 6.49 8.34e-11# get the coef
coef(logistic_model_goout_alcohol) %>% exp()## (Intercept) goout
## 0.03554946 2.13048020# get the exp of the coef
exp(coef(logistic_model_goout_alcohol))## (Intercept) goout
## 0.03554946 2.13048020# get the intervals
confint(logistic_model_goout_alcohol) %>% exp()## Waiting for profiling to be done...## 2.5 % 97.5 %
## (Intercept) 0.01515176 0.07850224
## goout 1.70566958 2.69518608# plots
data %>%
ggplot(aes(x= factor(high_use), y= goout)) +
geom_boxplot() +
labs(title = "Going out with friends vs the alcohol consumption",
x = "High consumption",
y = "go out with friends (from 1 to 5)")
# the students who are drinking more are more numerous among the students who gave the score 4 and 5 for the goout variable.
data %>%
ggplot(aes(x = factor(goout), fill = factor(high_use))) +
geom_bar(position = "fill") +
labs(
title = "Relationship Between Going Out Behavior and Alcohol Consumption",
x = "going out with friends from 1 to 5",
y = "Proportion of students",
fill = "High Use of alcohol: True of False"
)
# there are more score 1, 2 and 3 among the students who do not drink a lot
data %>%
ggplot(aes(x = factor(high_use), fill = factor(goout))) +
geom_bar(position = "fill") +
labs(
title = "Relationship Between Alcohol Consumption and going out behavior",
x = "High Use of alcohol: True of False",
y = "Proportion of students",
fill = "going out with friends from 1 to 5"
)
Hypothesis 3: students who study a lot have less risk to have a high consumption of alcohol
Comments about the below code chunk: The p value is very small and estimate is negative. We can conclude there is a negative relation between the study time and the alcohol consumption. When the study time increases the odds of high alcohol consumption decreases.
The coefficient for Studytime is -0.6208. The odds ratio associated with a one-unit change in Studytime is approximately exp(-0.6208) = 0.538. For a one-unit decrease in Studytime, the odds of ‘high_use’ decrease by about 1 - 0.538 = 46.2%.
With 95% confidence, the true value of the odds ratio for the intercept lies between approximately 0.79 and 2.67. For the study time, the odds ratio lies between 0.39 and 0.72, since it is < 1, there is a statistical significance with a negative effect of this variable on the high_use.
The residual deviance is smaller than the null deviance, it means the model explains the variable high use. The AIC is smaller than the first model but not as small as the second model.
logistic_model_studytime_alcohol <- glm(high_use ~ studytime, data = data, family = binomial)
summary(logistic_model_studytime_alcohol)##
## Call:
## glm(formula = high_use ~ studytime, family = binomial, data = data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.3649 0.3102 1.176 0.239
## studytime -0.6208 0.1542 -4.027 5.66e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.04 on 369 degrees of freedom
## Residual deviance: 433.82 on 368 degrees of freedom
## AIC: 437.82
##
## Number of Fisher Scoring iterations: 4tidy(logistic_model_studytime_alcohol)## # A tibble: 2 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 0.365 0.310 1.18 0.239
## 2 studytime -0.621 0.154 -4.03 0.0000566# get the coef
coef(logistic_model_studytime_alcohol) %>% exp()## (Intercept) studytime
## 1.4403849 0.5375316#Get the odds - exponential of coefficient:
exp(coef(logistic_model_studytime_alcohol))## (Intercept) studytime
## 1.4403849 0.5375316# get the intervals
confint(logistic_model_studytime_alcohol) %>% exp()## Waiting for profiling to be done...## 2.5 % 97.5 %
## (Intercept) 0.7881099 2.6654451
## studytime 0.3933727 0.7208174# This plot shows the difference in median and mean for the students who drink a lot or not
data %>%
ggplot(aes(x= factor(high_use), y= studytime)) +
geom_boxplot() +
labs(title = "Study time vs the alcohol consumption",
x = "High consumption",
y = "Study time (from 1 to 4)")
# way more students among students who score they study time around 1 and 2
data %>%
ggplot(aes(x = factor(studytime), fill = factor(high_use))) +
geom_bar(position = "fill") +
labs(
title = "Relationship Between Study time Behavior and Alcohol Consumption",
x = "Study time from 1 to 4",
y = "Proportion of students",
fill = "High Use of alcohol: True of False"
)
# way more score 1 and way less 3 and 4 among students who drink a lot.
data %>%
ggplot(aes(x = factor(high_use), fill = factor(studytime))) +
geom_bar(position = "fill") +
labs(
title = "Relationship Between Alcohol Consumption and Study time behavior",
x = "High Use of alcohol: True of False",
y = "Proportion of students",
fill = "Study time from 1 to 4"
)
Hypothesis 4: students who chose the school because of its reputation has less risk to have a high consumption of alcohol
Comments about the below code: The p value is 0,022 which tells that there is a correlation between the reason to choose a school and the consumption of alcohol.
We don’t know with the test which reason is the most chosen by the students who drink a lot of alcohol vs the ones who don’t. Some plots can help us identify it (check below).
# Creating a contingency table
chi_table_alcohol_reasons <- table((data$high_use), as.factor(data$reason))
# Performing the chi-squared test
chi_square_test <- chisq.test(chi_table_alcohol_reasons )
chi_square_test##
## Pearson's Chi-squared test
##
## data: chi_table_alcohol_reasons
## X-squared = 9.563, df = 3, p-value = 0.02267# Summary of the test
summary(chi_square_test)## Length Class Mode
## statistic 1 -none- numeric
## parameter 1 -none- numeric
## p.value 1 -none- numeric
## method 1 -none- character
## data.name 1 -none- character
## observed 8 table numeric
## expected 8 -none- numeric
## residuals 8 table numeric
## stdres 8 table numeric# This show that students who don't drink a lot have chosen their school more for its reputation than the one who drink a lot. For the ones who drink, the reasons home and other are way more important than for the students who don't drink too much. The reason courses seem to be equally important for them.
data %>%
ggplot(aes(x = high_use, fill = reason)) +
geom_bar(position = "fill") +
labs(
title = "Relationship Between Reason to chose their school and Alcohol Consumption",
x = "High Consumption of a",
y = "Proportion of students",
fill = "High Use of alcohol: True of False"
)
data %>%
ggplot(aes(x = reason, fill = high_use)) +
geom_bar(position = "fill") +
labs(
title = "Relationship Between Alcohol Consumption and Reason to chose their school",
x = "High Use of alcohol: True of False",
y = "Proportion of students",
fill = "Reason to choose the school"
)
Other models to test:
By adding each variable together, we can see that the variation in high use is better explained (this is because the deviance is decreasing from 452 to 379).
All three variables (goout, famrel, and studytime) have statistical significance in predicting alcohol consumption (high_use).
logistic_model_3var <- glm(high_use ~ goout + famrel + studytime, data = data, family = binomial)
logistic_model_3var_2 <- glm(high_use ~ goout * famrel * studytime, data = data, family = binomial)
summary(logistic_model_3var)##
## Call:
## glm(formula = high_use ~ goout + famrel + studytime, family = binomial,
## data = data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.7286 0.6941 -1.050 0.29381
## goout 0.7777 0.1203 6.467 9.99e-11 ***
## famrel -0.3777 0.1367 -2.763 0.00573 **
## studytime -0.6139 0.1672 -3.673 0.00024 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.04 on 369 degrees of freedom
## Residual deviance: 379.74 on 366 degrees of freedom
## AIC: 387.74
##
## Number of Fisher Scoring iterations: 4tidy(logistic_model_3var)## # A tibble: 4 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) -0.729 0.694 -1.05 2.94e- 1
## 2 goout 0.778 0.120 6.47 9.99e-11
## 3 famrel -0.378 0.137 -2.76 5.73e- 3
## 4 studytime -0.614 0.167 -3.67 2.40e- 4# Anova conclusion: The p-value associated with the difference in deviance (Pr(>Chi)) is 0.1362, suggesting that the difference in fit between the two models is not statistically significant
anova(logistic_model_3var, logistic_model_3var_2, test="LRT")## Analysis of Deviance Table
##
## Model 1: high_use ~ goout + famrel + studytime
## Model 2: high_use ~ goout * famrel * studytime
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 366 379.74
## 2 362 372.74 4 6.995 0.1362# when comparing with for instanc: logistic_model_family_alcohol, the residual deviance of logistic_model_3var is better and the p is significant.
anova(logistic_model_3var, logistic_model_family_alcohol, test="LRT")## Analysis of Deviance Table
##
## Model 1: high_use ~ goout + famrel + studytime
## Model 2: high_use ~ famrel
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 366 379.74
## 2 368 446.67 -2 -66.934 2.921e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# best model is the logistic_model_3var according to BIC and AIC, because metrics are smaller for this model
AIC(logistic_model_3var, logistic_model_3var_2)## df AIC
## logistic_model_3var 4 387.7372
## logistic_model_3var_2 8 388.7422BIC(logistic_model_3var, logistic_model_3var_2)## df BIC
## logistic_model_3var 4 403.3912
## logistic_model_3var_2 8 420.0502question 6: Model 1 (high_use ~ goout + famrel + studytime)
Using the variables which, according to your logistic regression model, had a statistical relationship with high/low alcohol consumption, explore the predictive power of you model. Provide a 2x2 cross tabulation of predictions versus the actual values and optionally display a graphic visualizing both the actual values and the predictions. Compute the total proportion of inaccurately classified individuals (= the training error) and comment on all the results. Compare the performance of the model with performance achieved by some simple guessing strategy.
# prediction based on the model: logistic_model_3var
predictions <- predict(logistic_model_3var, type = "response")
#Create the confusion matrix:
table(data$high_use, predictions > 0.5)##
## FALSE TRUE
## FALSE 235 24
## TRUE 63 48# correct are False false + true true = 235 + 48 = 293
# wrong are False True and True False = 24 + 63 = 87
# Probability of miss classifications = 87/(293+87) = 23%
# odds of miss classification = 87/293 = 30%
# precision: correctly positive among all positives predictions = 0.33%
precision <- 24 / (24 + 48)
# recall: correctly positive among all actual positives = 0.28%
recall <- 24 / (24 + 63)
# F1 score: 0.30%
2 * (precision * recall) / (precision + recall)## [1] 0.3018868 # Accuracy of always predicting the majority class: if the majority class is "FALSE," then predicting "FALSE" for every instance would result in a correct prediction rate of 70%
mean(data$high_use == FALSE)## [1] 0.7# Calculate the total proportion of errors is 23.5%. This means my model is a bit better than the majority class which is 70% (30% of non-accuracy).
mean(data$high_use != (predictions > 0.5))## [1] 0.2351351# to conclude, my model is a bit better than using the prediction based on the majority class. But it should be improved for better accuracy!Bonus:
I had to look at chat gpt fully for this one. I did not know the method. I now understand this method splits the data in 10 data samples and will use 9 of them to predict the last one, rotating to test them all. In that case, the average of prediction for the different tests is 74.9% which is better than the 70% (majority class).
library(caret)## Loading required package: lattice##
## Attaching package: 'caret'## The following object is masked from 'package:purrr':
##
## liftlibrary(lattice)
# Define the model formula
formula <- factor(high_use) ~ goout + famrel + studytime
# Define the number of folds for cross-validation
num_folds <- 10
# Create a training control with 10-fold cross-validation
train_control <- trainControl(method = "cv", number = num_folds)
# Train the model using 10-fold cross-validation
model <- train(formula, data = data, method = "glm", family = binomial, trControl = train_control)
# Get cross-validated results
results <- model$resample
str(results)## 'data.frame': 10 obs. of 3 variables:
## $ Accuracy: num 0.784 0.711 0.757 0.73 0.757 ...
## $ Kappa : num 0.454 0.205 0.368 0.353 0.33 ...
## $ Resample: chr "Fold01" "Fold02" "Fold03" "Fold04" ...# Calculate mean prediction error
mean(results$Accuracy)## [1] 0.7570887Chapter 4
Question 1, 2 and 3
dataset description:
The Boston research seems to look for correlations between the standard of living, housing quality, and other criteria such as the urban environment, accessibility, and demographics within the Boston area.
Each row represents data for a specific town and has information about various factors such as crime rate, proportion of non-retail business acres, property tax rates, pupil-teacher ratios, etc.
This data frame contains the following columns:
crim - per capita crime rate by town.
zn - proportion of residential land zoned for lots over 25,000 sq.ft.
indus - proportion of non-retail business acres per town.
chas - Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
nox - nitrogen oxides concentration (parts per 10 million).
rm - average number of rooms per dwelling.
age - proportion of owner-occupied units built prior to 1940.
dis - weighted mean of distances to five Boston employment centres.
rad - index of accessibility to radial highways.
tax - full-value property-tax rate per $10,000.
ptratio - pupil-teacher ratio by town.
black - 1000(Bk−0.63) square Bk is the proportion of blacks by town.
lstat - lower status of the population (percent).
medv - median value of owner-occupied homes in $1000s.
Data analyzes:
Explore the structure and the dimensions of the dataset. Show a graphical overview of the data and show summaries of the variables in the data. Describe and interpret the outputs, the distributions of the variables and the relationships between them.
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectlibrary(dplyr)
library(ggplot2)
library(GGally)
library(tidyr)
setwd("/Users/margot/Desktop/Desktop - MacBook Pro de MARGOT/Open data with R 2023/IODS-project")
Boston <- Boston
# explore the mean, median, min, max per variable
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00# 506 entries with 14 variables
dim(Boston)## [1] 506 14# Data type: all num or int
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...# let's create points plot for each variable. -> not easy to read as too many variables
pairs(Boston)
# Let's analyze the histogram for each variable. To use the key facet wrap that will draw one chart per key, we need to have a table with 2 columns: the keys which are all the variables and then the metric.
gather(Boston) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") +
geom_histogram()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Let's compute a correlation matrix
matrix_correlation_var <- cor(Boston)
# Visualize correlation matrix as a heatmap
library(reshape2)##
## Attaching package: 'reshape2'## The following object is masked from 'package:tidyr':
##
## smithsggplot(data = melt(matrix_correlation_var), aes(Var1, Var2, fill = value)) +
geom_tile() +
scale_fill_gradient2(low = "blue", high = "red", mid = "white", midpoint = 0)
according to the matrix - Negative relationships between certain variables:
lstat and medv - lower status of population and the median price of homes owned by occupants
lstat and rm - lower status of population and the average of room numbers per home
tax and medv - the percentage at which a property is taxed based on its assessed value and the the median price of homes owned by occupants
dis and lstat - the weighted average distance from each house to these employment centers and the lower status of population
dis and age - the weighted average distance from each house to these employment centers and the percentage or fraction of homes that are owner-occupied and were constructed before the year 1940.
dis and nox - the weighted average distance from each house to these employment centers and nitrogen oxides concentration.
dis and indus - the weighted average distance from each house to these employment centers and the proportion of non-retail business acres per town (manufacturing facilities, industrial parks, office spaces, warehouses etc).
tax and dis - the percentage at which a property is taxed based on its assessed value and the weighted average distance from each house to these employment centers
zn and age - the percentage or fraction of land within residential areas that is for individual lots with a minimum size of over 25,000 square feet and the percentage or fraction of homes that are owner-occupied and were constructed before the year 1940
zn and nox - the percentage or fraction of land within residential areas that is for individual lots with a minimum size of over 25,000 square feet and the nitrogen oxides concentration.
zn and indus - the percentage or fraction of land within residential areas that is for individual lots with a minimum size of over 25,000 square feet and the proportion of non-retail business acres per town (manufacturing facilities, industrial parks, office spaces, warehouses etc).
Positive relationships between variables
medv and rm - the median price of homes owned by occupants and the average of room numbers per home
tax and indus - the percentage at which a property is taxed based on its assessed value and the proportion of non-retail business acres per town.
tax and nox - the percentage at which a property is taxed based on its assessed value and the nitrogen oxides concentration.
age and indus - the percentage or fraction of homes that are owner-occupied and were constructed before the year 1940 and the proportion of non-retail business acres per town.
age and nox- the percentage or fraction of homes that are owner-occupied and were constructed before the year 1940 and the nitrogen oxides concentration.
nox and indus - the nitrogen oxides concentration and the proportion of non-retail business acres per town.
–> tests in my model: sltat, medv, rm, tax, dis, age, nox, indus, zn
Test some models based on the dataset and observed correlation
library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
##
## recode## The following object is masked from 'package:purrr':
##
## somemodel1 <- glm(lstat ~ medv + dis + rm,data=Boston)
model2 <- glm(medv ~ rm + tax + lstat ,data=Boston)
# all the VIF are between 1 and 2.1 which is ok. It suggest a low multicollinearity and imply that the variance of the estimated coefficients is not significantly inflated due to collinearity
vif(model1)## medv dis rm
## 1.982257 1.068810 1.940168vif(model2)## rm tax lstat
## 1.610953 1.426005 2.092901#let’s calculate the corresponding ODDS ratios and confidence intervals (95%):
OR1 <- coef(model1) %>% exp
OR2 <- coef(model2) %>% exp
CI1 <- confint(model1) %>% exp## Waiting for profiling to be done...CI2 <- confint(model2) %>% exp## Waiting for profiling to be done...# the confidence interval for an odds ratio doesn't span 1 = there's a statistically significant effect for both models
cbind(OR1, CI1) ## OR1 2.5 % 97.5 %
## (Intercept) 1.764803e+16 4.023272e+14 7.741285e+17
## medv 6.606608e-01 6.250484e-01 6.983022e-01
## dis 3.292580e-01 2.756488e-01 3.932934e-01
## rm 1.682751e-01 8.210665e-02 3.448749e-01cbind(OR2, CI2)## OR2 2.5 % 97.5 %
## (Intercept) 0.6073487 0.001289619 286.0319984
## rm 181.1868455 76.546116788 428.8744403
## tax 0.9935198 0.990167804 0.9968832
## lstat 0.5754723 0.522466602 0.6338557# the residual deviance is way smaller than the null deviance. It indicates a reasonably good fit of the model to the data.
summary(model1)##
## Call:
## glm(formula = lstat ~ medv + dis + rm, data = Boston)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 37.40940 1.92918 19.391 < 2e-16 ***
## medv -0.41451 0.02827 -14.662 < 2e-16 ***
## dis -1.11091 0.09067 -12.252 < 2e-16 ***
## rm -1.78215 0.36612 -4.868 1.51e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 17.22406)
##
## Null deviance: 25752.4 on 505 degrees of freedom
## Residual deviance: 8646.5 on 502 degrees of freedom
## AIC: 2882.2
##
## Number of Fisher Scoring iterations: 2# medv and rm also influences each other, so let's modify the model a bit
model11 <- glm(lstat ~ medv + dis + rm + rm * medv ,data=Boston)
summary(model11)##
## Call:
## glm(formula = lstat ~ medv + dis + rm + rm * medv, data = Boston)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 74.41052 3.64466 20.42 <2e-16 ***
## medv -1.94316 0.13517 -14.38 <2e-16 ***
## dis -0.89569 0.08285 -10.81 <2e-16 ***
## rm -7.55541 0.59817 -12.63 <2e-16 ***
## medv:rm 0.22526 0.01957 11.51 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 13.64918)
##
## Null deviance: 25752.4 on 505 degrees of freedom
## Residual deviance: 6838.2 on 501 degrees of freedom
## AIC: 2765.5
##
## Number of Fisher Scoring iterations: 2# same for this one, residual deviance is way smaller than the null deviance.
summary(model2)##
## Call:
## glm(formula = medv ~ rm + tax + lstat, data = Boston)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.498652 3.140239 -0.159 0.873895
## rm 5.199529 0.439618 11.827 < 2e-16 ***
## tax -0.006501 0.001724 -3.770 0.000182 ***
## lstat -0.552564 0.049302 -11.208 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 29.90866)
##
## Null deviance: 42716 on 505 degrees of freedom
## Residual deviance: 15014 on 502 degrees of freedom
## AIC: 3161.4
##
## Number of Fisher Scoring iterations: 2Question 4
Standardizing the data. Variables often have different scales and units, making direct comparisons challenging. Standardization brings all variables to a common scale, allowing for fair comparisons between different variables. It makes the distribution of each variable more consistent, with a mean of 0 and a standard deviation of 1. This normalization aids in interpreting coefficients and comparing the relative importance of different predictors in regression. Standardizing ensures that each variable contributes equally to these techniques, preventing one variable from dominating the analysis due to its scale. It allows easier comparison of the magnitude of the effect of each variable on the outcome. Finally it can mitigate issues related to multicollinearity in regression analysis by putting variables on a similar scale, reducing the impact of differing scales on regression coefficients.
# select numerical values
Boston_numeric_cols <- Boston[, sapply(Boston, is.numeric)]
# The scale() to standardize and transform the data to have a mean of 0 and a standard deviation of 1.
scaled_boston <- scale(Boston_numeric_cols)
# convert to a data frame
scaled_table_boston <- as.data.frame(scaled_boston)
# how did the data change? Mean is now 0 so it has worked;
summary(scaled_table_boston)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# use the cut function to create categorical variables based on intervals or breaks in a numerical variable. We do this process for the crim variable from 0 to 0.25 to 0.50 to 0.75 to 1 (quantiles). Add labels for each category.
# include.lowest = TRUE is to ensure there is no NA category.
quantiles <- quantile(Boston$crim, probs = seq(0, 1, by = 0.25), na.rm = TRUE)
interval_labels <- c("premier_quantil", "second_quantil", "third_quantil", "fourth_quantil")
scaled_table_boston$quantiles_crime <- cut(Boston$crim, quantiles, labels= interval_labels,include.lowest = TRUE)
# Notes: Quantiles derived from numeric values can be considered categorical or continuous. When quantiles represent discrete categories or bins that divide a continuous variable into distinct groups, they are treated as categorical (e. small, medium, big). If quantiles represent numeric values that indicate the position or value relative to the distribution of a continuous variable (e.g., the 25th percentile, median, 75th percentile), they are considered continuous.
# drop the former column crim and create a new table.
Boston_new <- scaled_table_boston %>% dplyr::select(-crim)
# We need 80% of the rows from total rows
train_size <- round(0.8 * nrow(Boston_new))
# Select a sample randomly among the dataset 80%
train_set <- sample(seq_len(nrow(Boston_new)), size = train_size)
# Create training and testing subsets
train_data <- Boston_new[train_set, ]
test_data <- Boston_new[-train_set, ]Question 5:
Fit the linear discriminant analysis on the train set. Use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables. Draw the LDA (bi)plot.
Linear Discriminant Analysis (LDA) can be quite useful in various real-life scenarios where classification or dimensionality reduction is required such as: Marketing and Customer Segmentation, product quality control, credit scoring.
It is ideal when we have multiple variables that can help us categorize each entry in one specific group where all others will have similar mean, and variance. The goal is to predict the categorical outcome or class based on a set of predictors.
LDA transforms the original variables into a smaller set of new variables, known as linear discriminants. These new variables (linear discriminants) are created in a way that maximizes the separation between different categories or classes in the data.By using the information captured in the linear discriminants, LDA helps in accurately assigning new observations (data points) to their respective categories or classes. LDA takes variables, combines them in a smart way to create new variables that are helpful for understanding differences between groups or categories, and then uses this understanding to categorize or classify new data based on what it has learned. This makes it useful for both simplifying data and making predictions about categories in that data.
LDA calculates the mean and variance for each predictor within each class or category in the target variable. It finds coefficients for these combinations by considering differences in means between classes and assumes equal variances for each predictor within classes. It assumes that the data approximately follows a normal distribution.
# check the impact of each variable on a categorical variable (quantiles_crime)
# each quantiles are approximately equal 25%
# LD1 explains 96% of the model. When LD1 captures 96% of the variance, it suggests that LD1 effectively separates the classes or groups in the data based on their mean differences and variance. Within LD1, data points are grouped together in a way that maximizes the separation between classes while minimizing the variation within each class.
# Positive coefficients indicate that higher values of that variable contribute to a higher score in that particular discriminant, while negative coefficients suggest the opposite. The larger the magnitude of the coefficient, the greater the impact of that variable on the discriminant.
library(MASS)
# Fit LDA on the train set - LDA is a statistical method used for classification. It fits an LDA model using the train_data, where quantiles_crime is the target variable to predict based on the other variables
lda_model <- lda(quantiles_crime ~ ., data = train_data)
# Predict on the test set by using the function predict to apply the model created from the train_data to test_data -> the idea here is to predict the class labels for this test_data
lda_pred <- predict(lda_model, test_data)$class # class extract the predicted class labels
# actual crime quantiles
actual_crime_categories <- test_data$quantiles_crime
# Extract LD scores from the LDA model's predictions
lda_scores <- predict(lda_model, test_data)$x # x accesses the matrix of posterior probabilities or scores associated with each class for the observations in the test_data.
# Create a dataframe with LD scores from first 2 columns of the lda_score and the predicted classes. Combining LD1, LD2, and the predicted classes for visualization. In many cases, visualizing beyond LD1 and LD2 might become complex to display effectively in two-dimensional plots. LD1 and LD2 are typically chosen for visualization as they capture the most discrimination power while allowing for a clearer visualization on a 2D plot.
plot_data <- data.frame(LD1 = lda_scores[, 1], LD2 = lda_scores[, 2], prediction_crime_quantile = as.factor(lda_pred))
plot_data## LD1 LD2 prediction_crime_quantile
## 13 -2.19495906 0.05903739 second_quantil
## 20 -1.98773503 -0.52094792 second_quantil
## 21 -1.49908173 -0.95224424 third_quantil
## 24 -1.56564250 -1.06178847 third_quantil
## 39 -2.21004108 0.15120955 second_quantil
## 46 -3.30338408 0.24096910 second_quantil
## 49 -2.22126948 -0.95126769 second_quantil
## 61 -0.93282684 0.90823135 second_quantil
## 64 -1.29234733 0.76811854 second_quantil
## 65 -3.12145993 -0.34896538 second_quantil
## 74 -3.29156028 0.47978327 second_quantil
## 82 -2.62520888 0.56228796 premier_quantil
## 86 -3.10163904 -0.24846952 second_quantil
## 92 -3.16892206 -0.43989685 second_quantil
## 93 -2.64192556 0.58322356 premier_quantil
## 98 -3.31473562 -1.09742740 second_quantil
## 101 -1.57223862 -0.86169268 third_quantil
## 106 -1.23809842 -0.72007431 third_quantil
## 107 -1.22462680 -0.72834052 third_quantil
## 118 -1.34645502 -0.54799558 third_quantil
## 133 -1.33540967 -2.01026803 third_quantil
## 138 -1.38233763 -1.86481790 third_quantil
## 148 0.06002818 -2.77823674 third_quantil
## 152 -0.24758231 -2.68662138 third_quantil
## 155 -0.65994332 -2.95090625 third_quantil
## 156 -0.47729143 -2.75626171 third_quantil
## 162 -1.38267474 -2.41761081 third_quantil
## 166 -1.28786330 -1.54996219 third_quantil
## 167 -1.34089136 -2.64123224 third_quantil
## 168 -1.32960771 -1.36894830 third_quantil
## 175 -1.99257815 -0.05827807 second_quantil
## 179 -2.01173128 -0.54740105 second_quantil
## 181 -2.56727400 -1.20727137 third_quantil
## 186 -2.52083834 -0.54620382 second_quantil
## 194 -4.41841995 2.04956409 premier_quantil
## 212 -2.02305826 -1.03071934 second_quantil
## 214 -2.64258917 -0.29272399 second_quantil
## 215 -2.17042542 -0.16114623 second_quantil
## 222 -0.68657531 -0.64490619 second_quantil
## 224 -0.78950217 -0.37098639 third_quantil
## 234 -0.82052112 -1.16624410 third_quantil
## 251 -2.12718479 1.24941152 premier_quantil
## 257 -3.22605746 2.43809473 premier_quantil
## 259 -1.30344843 -1.07330538 third_quantil
## 260 -1.39975619 -0.75311722 third_quantil
## 266 -1.61434309 0.14555608 third_quantil
## 267 -1.32953803 -0.84632984 third_quantil
## 270 -3.00405224 0.23746305 second_quantil
## 280 -2.70117177 0.82659369 premier_quantil
## 283 -2.74442403 -0.21435174 second_quantil
## 286 -4.37411298 1.85089945 premier_quantil
## 294 -3.29708997 0.08945746 second_quantil
## 297 -3.01949543 -0.54309744 second_quantil
## 301 -2.78704951 2.34421790 premier_quantil
## 308 -1.27478332 1.13087979 premier_quantil
## 309 -2.25953966 -0.86039721 third_quantil
## 320 -2.22669990 -0.69386006 second_quantil
## 322 -2.19314060 -0.20411626 second_quantil
## 325 -2.31673625 -0.10149749 second_quantil
## 330 -3.15016061 0.34245714 second_quantil
## 338 -1.97067647 -0.24500701 second_quantil
## 339 -2.20264295 0.07876816 second_quantil
## 340 -2.11654735 0.01513690 second_quantil
## 348 -2.94349176 2.76843746 premier_quantil
## 349 -2.98679641 2.63024741 premier_quantil
## 353 -3.15770834 2.18893191 premier_quantil
## 356 -2.81926788 2.81953007 premier_quantil
## 360 6.28475167 -0.21754982 fourth_quantil
## 366 6.59078774 0.59263634 fourth_quantil
## 367 6.51793327 0.24068984 fourth_quantil
## 375 6.92428804 0.27931281 fourth_quantil
## 376 5.73510806 0.22114940 fourth_quantil
## 378 6.00737853 0.16219149 fourth_quantil
## 380 6.05947758 0.35744068 fourth_quantil
## 386 6.48573099 0.28563022 fourth_quantil
## 390 6.28583386 0.33607651 fourth_quantil
## 402 6.00576993 0.33966215 fourth_quantil
## 403 6.13729073 0.14687884 fourth_quantil
## 410 6.33183637 0.03242766 fourth_quantil
## 411 6.21645960 0.95188971 fourth_quantil
## 415 7.22526153 0.24293928 fourth_quantil
## 422 6.24957653 0.17896627 fourth_quantil
## 437 6.63743599 0.04886889 fourth_quantil
## 439 7.03548322 -0.13812762 fourth_quantil
## 450 6.28310020 -0.01419889 fourth_quantil
## 452 6.16902774 -0.11278942 fourth_quantil
## 453 6.12856747 0.02544060 fourth_quantil
## 454 6.05873291 -0.35605841 fourth_quantil
## 470 5.36955703 1.15572044 fourth_quantil
## 471 5.59006042 0.65749835 fourth_quantil
## 477 5.77328416 0.42689040 fourth_quantil
## 482 5.06944873 0.96145356 fourth_quantil
## 486 5.13094364 1.00860904 fourth_quantil
## 489 -1.30568868 -1.87826668 third_quantil
## 493 -1.50897155 -1.95177955 third_quantil
## 498 -1.10881911 -0.61579729 third_quantil
## 499 -1.15127139 -0.60799991 third_quantil
## 500 -1.02168534 -0.54515001 third_quantil
## 501 -1.07827967 -0.68616699 third_quantil
## 503 -2.92111074 -1.18254661 third_quantil
## 506 -3.10533674 -0.90557694 premier_quantil# Create a scatterplot of LD1 and LD2
plot_LDA <- ggplot(plot_data, aes(x = LD1, y = LD2, color = prediction_crime_quantile)) +
geom_point() +
labs(title = "LDA Biplot with Predicted Crime Quantiles")
plot_LDA
# adding real values - comparison of actual vs predicted values in test_data
realVsPred_plot <- plot_LDA +
geom_point(aes(color = actual_crime_categories), size = 4, alpha = 0.1) +
labs(color = "Real Quantiles of Crime")
realVsPred_plot
# the accuracy of predictions using test data
accuracy <- mean(lda_pred == test_data$quantiles_crime)
print(paste("Accuracy of LDA model on test data:", round(accuracy * 100, 2), "%"))## [1] "Accuracy of LDA model on test data: 68.32 %"Question 6:
Save the crime categories from the test set and then remove the categorical crime variable from the test dataset. Then predict the classes with the LDA model on the test data. Cross tabulate the results with the crime categories from the test set. Comment on the results. (0-3 points)
# save the crime data:
actual_crime_categories <- test_data$quantiles_crime
# Remove the categorical crime variable from the test dataset
test_data_without_crime <- test_data %>% dplyr::select(-quantiles_crime)
# get the classes based on the model - this was done earlier with lda_pred. so I am a bit confused.
lda_pred_test <- predict(lda_model, test_data_without_crime)$class
# get the table with the prediction vs actual - results are same between the 2 ways since I did 2 times the same steps. I might have missed something in the requests.
cross_tab <- table(Predicted = lda_pred_test, Actual = actual_crime_categories)
cross_tab## Actual
## Predicted premier_quantil second_quantil third_quantil fourth_quantil
## premier_quantil 11 3 0 0
## second_quantil 11 13 6 0
## third_quantil 2 9 20 0
## fourth_quantil 0 0 1 25cross_table <- table(Predicted = lda_pred, Actual = actual_crime_categories)
cross_table## Actual
## Predicted premier_quantil second_quantil third_quantil fourth_quantil
## premier_quantil 11 3 0 0
## second_quantil 11 13 6 0
## third_quantil 2 9 20 0
## fourth_quantil 0 0 1 25Question 7:
Reload the Boston dataset and standardize the dataset (we did not do this in the Exercise Set, but you should scale the variables to get comparable distances). Calculate the distances between the observations. Run k-means algorithm on the dataset. Investigate what is the optimal number of clusters and run the algorithm again. Visualize the clusters (for example with the pairs() or ggpairs() functions, where the clusters are separated with colors) and interpret the results. (0-4 points)
# Boston is reload
Boston_load <- Boston
# scale of Boston
Boston_scaled <- scale(Boston_load)
# Calculate distance betwwen scaled data points:
distances <- dist(Boston_scaled)
# Visualize the distances using fviz_dist()
library(factoextra)## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBafviz_dist(distances, gradient = list(low = "#00AFBB", mid = "white", high = "#FC4E07"))
# k mean: The k-means algorithm groups observations into clusters based on their similarity, aiming to minimize the variation within clusters while maximizing the variation between clusters. We need to define the number of clusters we need to do the kmean analyse.
# Elbow Method to find optimal number of clusters: Calculate the Within-Cluster-Sum of Squared Errors (WSS) for different values of k, and choose the k for which WSS becomes first starts to diminish. In the plot of WSS-versus-k, this is visible as an elbow.
# The Squared Error for each point is the square of the distance of the point from its representation i.e. its predicted cluster center.
# The WSS score is the sum of these Squared Errors for all the points.
wss <- numeric(10) # Initialize within-cluster sum of squares vector
for (i in 1:10) {
kmeans_model <- kmeans(Boston_scaled, centers = i, nstart = 10)
wss[i] <- kmeans_model$tot.withinss # Store within-cluster sum of squares
}
# Plotting the Elbow Method
plot(1:10, wss, type = "b", xlab = "Number of Clusters", ylab = "Within-cluster Sum of Squares", main = "Elbow Method")
# or we can use the direct function for the Elbow method
kmeans_model_optimal2 <- fviz_nbclust(Boston_scaled, kmeans, method = "wss")
# or we can use the silhuette method
fviz_nbclust(Boston_scaled, kmeans, method = "silhouette")
# Visualize clusters using pairs() or ggpairs()
pairs(Boston_scaled, col = kmeans_model$cluster)
k <- 2 # 2 seems to be the best option according to the Elbow Method and the silhouette method
#Kmean model:
kmeans_model <- kmeans(Boston_scaled, centers = k, nstart = 25)
cluster_assignments <- kmeans_model$cluster
# visualize the clusters thanks to fviz_cluster function
fviz_cluster(kmeans_model, data = Boston_scaled)
library(GGally)
# Combine the scaled data with the cluster assignments (cluster 1 or 2)
clustered_data <- cbind(as.data.frame(Boston_scaled), Cluster = as.factor(cluster_assignments))
# Visualize clusters using ggpairs()
ggpairs(clustered_data, aes(color = Cluster))## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero
## Warning in cor(x, y): the standard deviation is zero## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# Mean for each cluster and variable:
clustered_data %>%
mutate(Cluster = kmeans_model$cluster) %>%
group_by(Cluster) %>%
summarise_all("mean")## # A tibble: 2 × 15
## Cluster crim zn indus chas nox rm age dis rad
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -0.389 0.262 -0.615 0.00291 -0.582 0.245 -0.433 0.454 -0.583
## 2 2 0.724 -0.487 1.14 -0.00541 1.08 -0.455 0.805 -0.844 1.08
## # ℹ 5 more variables: tax <dbl>, ptratio <dbl>, black <dbl>, lstat <dbl>,
## # medv <dbl>model_predictors <- train_data %>% dplyr::select(-quantiles_crime)
# check the dimensions
dim(model_predictors)## [1] 405 13dim(lda_model$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda_model$scaling
matrix_product <- as.data.frame(matrix_product)
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ordinal':
##
## slice## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutplot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')Chapter 5
Question 1
Move the country names to rownames (see Exercise 5.5). Show a graphical overview of the data and show summaries of the variables in the data. Describe and interpret the outputs, commenting on the distributions of the variables and the relationships between them. (0-3 points)
Data import and wrangling
library(tibble)
library(readr)
library(corrplot)## corrplot 0.92 loadedlibrary(ggplot2)
library(tidyr)
library(GGally)
human_new <- read_csv("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/human2.csv")## Rows: 155 Columns: 9## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (1): Country
## dbl (8): Edu2.FM, Labo.FM, Life.Exp, Edu.Exp, GNI, Mat.Mor, Ado.Birth, Parli.F
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# no NA
any(is.na(human_new))## [1] FALSE# analyze and summary of variables
str(human_new)## spc_tbl_ [155 × 9] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
## $ Country : chr [1:155] "Norway" "Australia" "Switzerland" "Denmark" ...
## $ Edu2.FM : num [1:155] 1.007 0.997 0.983 0.989 0.969 ...
## $ Labo.FM : num [1:155] 0.891 0.819 0.825 0.884 0.829 ...
## $ Life.Exp : num [1:155] 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ Edu.Exp : num [1:155] 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ GNI : num [1:155] 64992 42261 56431 44025 45435 ...
## $ Mat.Mor : num [1:155] 4 6 6 5 6 7 9 28 11 8 ...
## $ Ado.Birth: num [1:155] 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ Parli.F : num [1:155] 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...
## - attr(*, "spec")=
## .. cols(
## .. Country = col_character(),
## .. Edu2.FM = col_double(),
## .. Labo.FM = col_double(),
## .. Life.Exp = col_double(),
## .. Edu.Exp = col_double(),
## .. GNI = col_double(),
## .. Mat.Mor = col_double(),
## .. Ado.Birth = col_double(),
## .. Parli.F = col_double()
## .. )
## - attr(*, "problems")=<externalptr>summary(human_new)## Country Edu2.FM Labo.FM Life.Exp
## Length:155 Min. :0.1717 Min. :0.1857 Min. :49.00
## Class :character 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:66.30
## Mode :character Median :0.9375 Median :0.7535 Median :74.20
## Mean :0.8529 Mean :0.7074 Mean :71.65
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:77.25
## Max. :1.4967 Max. :1.0380 Max. :83.50
## Edu.Exp GNI Mat.Mor Ado.Birth
## Min. : 5.40 Min. : 581 Min. : 1.0 Min. : 0.60
## 1st Qu.:11.25 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65
## Median :13.50 Median : 12040 Median : 49.0 Median : 33.60
## Mean :13.18 Mean : 17628 Mean : 149.1 Mean : 47.16
## 3rd Qu.:15.20 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95
## Max. :20.20 Max. :123124 Max. :1100.0 Max. :204.80
## Parli.F
## Min. : 0.00
## 1st Qu.:12.40
## Median :19.30
## Mean :20.91
## 3rd Qu.:27.95
## Max. :57.50# Country is as rownames
human_new <- column_to_rownames(human_new, "Country")Graphical overview
# histogram per variable
pivot_longer(human_new, cols = everything()) %>%
ggplot(aes(value)) + facet_wrap("name", scales = "free") +
geom_histogram()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
# charts and correlations
ggpairs(human_new,lower=list(combo=wrap("facethist",binwidth=0.1)))
# correlation between variables
c <- cor(human_new)
corrplot(c, method="circle")
Commenting distributions and relationships between them
In majority, the distribution of these variables appears to deviate from a normal distribution.
Life expectancy, Education F/M, Education exp, GNI are negatively correlated with Maternal mortality and Adolescent birth.
Education F/M is positively correlated with Life expectancy, education exp, and GNI. Life expectancy is positively correlated with Education exp, and GNI. Education exp is also positively correlated with GNI. Maternal mortality is correlated to adolescent birth.
> This observed correlations appears coherent, highlighting logical relationships between the various variables.
Question 2
Perform principal component analysis (PCA) on the raw (non-standardized) human data. Show the variability captured by the principal components. Draw a biplot displaying the observations by the first two principal components (PC1 coordinate in x-axis, PC2 coordinate in y-axis), along with arrows representing the original variables. (0-2 points)
Non-standardized PCA
# PCA
pca_human <- prcomp(human_new)
# draw a biplot of the principal component representation and the original variables
biplot(pca_human, choices = 1:2)## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
# GNI has too big variances and is dominating the pca. The magnitude is too different between the metricsQuestion 3 - standardized PCA
Standardize the variables in the human data and repeat the above analysis. Interpret the results of both analysis (with and without standardizing). Are the results different? Why or why not? Include captions (brief descriptions) in your plots where you describe the results by using not just your variable names, but the actual phenomena they relate to. (0-4 points)
Scaled data & standardized PCA
# standardization of human_new
scaled_human <- scale(human_new)
# new PCA with scaled data
pca_human2 <- prcomp(scaled_human)
# draw a biplot of the PCA, adding captions that describe the phenomena
biplot(pca_human2, choices = 1:2,
xlab="PC1 - Country Socio-Economic Development", ylab="PC2 - Female Empowerment and Socio-Economic Development by Women", main = " Principal Component Analysis of Socio-Economic Indicators: Mapping Country Development Factors and Female Empowerment")
Include captions and descriptions on PCA:
PC1 - Country Socio-Economic Development. This component shows multiple socio-economic factors contributing to overall country development. It includes on one side, variables like Gross National Income, Life Expectancy, Education, and on the other side, variables like Maternal mortality and adolescent birth rate. This axis reflects a broad aspect of a country’s overall progress and development.
PC2 - Female Empowerment and Socio-Economic Development by Women. This component specifically highlights aspects related to female empowerment, participation (work or parli), and their impact on socio-economic development.
Question 4 - interpret the PCA
Give your personal interpretations of the first two principal component dimensions based on the biplot drawn after PCA on the standardized human data. (0-2 points):
In the first PCA, the GNI variable has too high metrics and too big variances vs the other variables. This variable dominates the PCA and therefore the PCA cannot be analyzed properly. The magnitude is too different between the metrics to compare them together without scaling them. After scaling the dataset, I did the PCA again to compare them more equally.
Analyze the standardized PCA: The arrows aligned with PC1 and its associated variables provide insight into the relationships between the variables themselves, it is the same for the PC2. The direction of the arrow suggests that the pointed variables have a significant and positive influence on the principal components represented in the plot. The direction of the arrow indicates which way the variable is correlated with the principal component. The length of the arrow represents the strength of the variable’s contribution to that principal component. Longer arrows indicate a stronger influence on that component.
On the left side of the chart, countries appear more developed, characterized by higher values in key development indicators such as Gross National Income, Life Expectancy at Birth, Life Expectancy in Education. These variables has significant influence, suggesting that the variables impact positively each other. As these metrics increase, the level of development tends to be higher. We can find in this subgroup many developed Western and Asian countries (e.g Switzerland, United States, Italy, Australia, Japan, Denmark,etc.). Some of them also have the characteristics of high labor rate of Female vs male which indicates a better situation for Women (e.g Iceland, Norway, Finland Sweden, Denmark). Conversely, countries like Qatar, Bahrain, UAE, and Saudi Arabia display a relatively good development level but possess lower female labor and parliament participation, impacting the overall women development and freedom. Some countries in the middle or even on the right of the PC1 do particularly well with the metrics about women development and freedom, such as Bolivia, Namibia, Burundi.
On the opposite side of the chart, variables such as Adolescent Birth Rate, Maternal Mortality Ratio, portray countries facing more challenging situations. Higher values in these variables correlate with less favorable conditions in a country. These countries, primarily from the Global South, such as Central African Republic, Congo, Burkina Faso face lower overall development. Women situations also vary based on their positioning along PC1, reflecting the diverse and unequal experiences of women in different countries (Afghanistan appears to have a worst situation for Women than in Zimbabwe for instance).
Question 5
Tea data import and exploration
The tea data comes from the FactoMineR package and it is measured with a questionnaire on tea: 300 individuals were asked how they drink tea (18 questions) and what are their product’s perception (12 questions).
In addition, some personal details were asked (4 questions). Load the tea dataset and convert its character variables to factors: tea <- read.csv(“https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/tea.csv”, stringsAsFactors = TRUE)
Explore the data briefly: look at the structure and the dimensions of the data. Use View(tea) to browse its contents, and visualize the data.
tea_new <- read.csv("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/tea.csv", stringsAsFactors = TRUE)
# View and summary of the dataset
view(tea_new)
str(tea_new)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "+60","15-24",..: 4 5 5 2 5 2 4 4 4 4 ...
## $ frequency : Factor w/ 4 levels "+2/day","1 to 2/week",..: 3 3 1 3 1 3 4 2 1 1 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...summary(tea_new)## breakfast tea.time evening lunch
## breakfast :144 Not.tea time:131 evening :103 lunch : 44
## Not.breakfast:156 tea time :169 Not.evening:197 Not.lunch:256
##
##
##
##
##
## dinner always home work
## dinner : 21 always :103 home :291 Not.work:213
## Not.dinner:279 Not.always:197 Not.home: 9 work : 87
##
##
##
##
##
## tearoom friends resto pub
## Not.tearoom:242 friends :196 Not.resto:221 Not.pub:237
## tearoom : 58 Not.friends:104 resto : 79 pub : 63
##
##
##
##
##
## Tea How sugar how
## black : 74 alone:195 No.sugar:155 tea bag :170
## Earl Grey:193 lemon: 33 sugar :145 tea bag+unpackaged: 94
## green : 33 milk : 63 unpackaged : 36
## other: 9
##
##
##
## where price age sex
## chain store :192 p_branded : 95 Min. :15.00 F:178
## chain store+tea shop: 78 p_cheap : 7 1st Qu.:23.00 M:122
## tea shop : 30 p_private label: 21 Median :32.00
## p_unknown : 12 Mean :37.05
## p_upscale : 53 3rd Qu.:48.00
## p_variable :112 Max. :90.00
##
## SPC Sport age_Q frequency
## employee :59 Not.sportsman:121 +60 :38 +2/day :127
## middle :40 sportsman :179 15-24:92 1 to 2/week: 44
## non-worker :64 25-34:69 1/day : 95
## other worker:20 35-44:40 3 to 6/week: 34
## senior :35 45-59:61
## student :70
## workman :12
## escape.exoticism spirituality healthy
## escape-exoticism :142 Not.spirituality:206 healthy :210
## Not.escape-exoticism:158 spirituality : 94 Not.healthy: 90
##
##
##
##
##
## diuretic friendliness iron.absorption
## diuretic :174 friendliness :242 iron absorption : 31
## Not.diuretic:126 Not.friendliness: 58 Not.iron absorption:269
##
##
##
##
##
## feminine sophisticated slimming exciting
## feminine :129 Not.sophisticated: 85 No.slimming:255 exciting :116
## Not.feminine:171 sophisticated :215 slimming : 45 No.exciting:184
##
##
##
##
##
## relaxing effect.on.health
## No.relaxing:113 effect on health : 66
## relaxing :187 No.effect on health:234
##
##
##
##
## # variables are factorial, except few ones. let's map the different categories thanks to the MCA
# But first let's keep only the categorical variables. An alternative would have been to change the non-catgeorical variables to categorical ones.
tea_new_2 <- tea_new %>% dplyr::select(-breakfast,-age)Multiple Correspondence Analysis (MCA)
On the tea data (or on just certain columns of the data, it is up to you!). Interpret the results of the MCA and draw at least the variable biplot of the analysis. You can also explore other plotting options for MCA. Comment on the output of the plots. (0-4 points)
library(FactoMineR)
# MCA
mca <- MCA(tea_new_2, graph = FALSE)
mca## **Results of the Multiple Correspondence Analysis (MCA)**
## The analysis was performed on 300 individuals, described by 34 variables
## *The results are available in the following objects:
##
## name description
## 1 "$eig" "eigenvalues"
## 2 "$var" "results for the variables"
## 3 "$var$coord" "coord. of the categories"
## 4 "$var$cos2" "cos2 for the categories"
## 5 "$var$contrib" "contributions of the categories"
## 6 "$var$v.test" "v-test for the categories"
## 7 "$var$eta2" "coord. of variables"
## 8 "$ind" "results for the individuals"
## 9 "$ind$coord" "coord. for the individuals"
## 10 "$ind$cos2" "cos2 for the individuals"
## 11 "$ind$contrib" "contributions of the individuals"
## 12 "$call" "intermediate results"
## 13 "$call$marge.col" "weights of columns"
## 14 "$call$marge.li" "weights of rows"# summary of the MCA
summary(mca)##
## Call:
## MCA(X = tea_new_2, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6 Dim.7
## Variance 0.092 0.084 0.072 0.060 0.057 0.055 0.050
## % of var. 5.913 5.392 4.629 3.881 3.672 3.502 3.188
## Cumulative % of var. 5.913 11.305 15.934 19.815 23.488 26.989 30.177
## Dim.8 Dim.9 Dim.10 Dim.11 Dim.12 Dim.13 Dim.14
## Variance 0.049 0.048 0.045 0.042 0.042 0.039 0.038
## % of var. 3.138 3.050 2.885 2.675 2.672 2.491 2.424
## Cumulative % of var. 33.315 36.365 39.251 41.926 44.598 47.089 49.513
## Dim.15 Dim.16 Dim.17 Dim.18 Dim.19 Dim.20 Dim.21
## Variance 0.036 0.036 0.034 0.032 0.032 0.031 0.030
## % of var. 2.320 2.305 2.173 2.077 2.049 2.016 1.944
## Cumulative % of var. 51.833 54.138 56.311 58.387 60.436 62.453 64.397
## Dim.22 Dim.23 Dim.24 Dim.25 Dim.26 Dim.27 Dim.28
## Variance 0.029 0.027 0.026 0.026 0.025 0.024 0.024
## % of var. 1.880 1.761 1.692 1.669 1.624 1.553 1.544
## Cumulative % of var. 66.276 68.038 69.730 71.399 73.023 74.577 76.120
## Dim.29 Dim.30 Dim.31 Dim.32 Dim.33 Dim.34 Dim.35
## Variance 0.023 0.022 0.022 0.021 0.020 0.019 0.019
## % of var. 1.456 1.426 1.404 1.339 1.264 1.246 1.221
## Cumulative % of var. 77.576 79.002 80.406 81.745 83.009 84.255 85.476
## Dim.36 Dim.37 Dim.38 Dim.39 Dim.40 Dim.41 Dim.42
## Variance 0.017 0.017 0.017 0.016 0.016 0.015 0.015
## % of var. 1.120 1.112 1.078 1.028 1.005 0.968 0.939
## Cumulative % of var. 86.596 87.708 88.786 89.814 90.819 91.786 92.725
## Dim.43 Dim.44 Dim.45 Dim.46 Dim.47 Dim.48 Dim.49
## Variance 0.013 0.013 0.012 0.012 0.011 0.010 0.010
## % of var. 0.866 0.837 0.802 0.739 0.697 0.674 0.666
## Cumulative % of var. 93.590 94.427 95.229 95.968 96.666 97.339 98.005
## Dim.50 Dim.51 Dim.52 Dim.53
## Variance 0.009 0.008 0.007 0.006
## % of var. 0.603 0.531 0.458 0.403
## Cumulative % of var. 98.608 99.139 99.597 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3 ctr
## 1 | -0.629 1.432 0.202 | 0.148 0.086 0.011 | 0.109 0.055
## 2 | -0.427 0.660 0.139 | 0.288 0.330 0.064 | -0.110 0.056
## 3 | 0.097 0.034 0.006 | -0.156 0.096 0.015 | 0.125 0.073
## 4 | -0.560 1.133 0.227 | -0.279 0.308 0.056 | -0.026 0.003
## 5 | -0.173 0.108 0.028 | -0.149 0.088 0.020 | 0.027 0.003
## 6 | -0.681 1.675 0.272 | -0.293 0.340 0.050 | -0.008 0.000
## 7 | -0.223 0.181 0.037 | 0.015 0.001 0.000 | 0.183 0.155
## 8 | -0.031 0.003 0.001 | 0.111 0.048 0.009 | -0.097 0.043
## 9 | -0.063 0.014 0.003 | 0.266 0.281 0.048 | 0.388 0.697
## 10 | 0.177 0.114 0.021 | 0.369 0.539 0.090 | 0.315 0.459
## cos2
## 1 0.006 |
## 2 0.009 |
## 3 0.009 |
## 4 0.000 |
## 5 0.001 |
## 6 0.000 |
## 7 0.025 |
## 8 0.007 |
## 9 0.103 |
## 10 0.066 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr cos2 v.test
## Not.tea time | -0.552 4.238 0.236 -8.396 | 0.001 0.000 0.000 0.019 |
## tea time | 0.428 3.285 0.236 8.396 | -0.001 0.000 0.000 -0.019 |
## evening | 0.296 0.961 0.046 3.704 | -0.404 1.961 0.085 -5.051 |
## Not.evening | -0.155 0.503 0.046 -3.704 | 0.211 1.025 0.085 5.051 |
## lunch | 0.590 1.630 0.060 4.231 | -0.406 0.844 0.028 -2.907 |
## Not.lunch | -0.101 0.280 0.060 -4.231 | 0.070 0.145 0.028 2.907 |
## dinner | -1.047 2.447 0.082 -4.965 | -0.080 0.016 0.000 -0.379 |
## Not.dinner | 0.079 0.184 0.082 4.965 | 0.006 0.001 0.000 0.379 |
## always | 0.342 1.281 0.061 4.276 | -0.255 0.784 0.034 -3.193 |
## Not.always | -0.179 0.670 0.061 -4.276 | 0.134 0.410 0.034 3.193 |
## Dim.3 ctr cos2 v.test
## Not.tea time 0.063 0.071 0.003 0.959 |
## tea time -0.049 0.055 0.003 -0.959 |
## evening 0.324 1.469 0.055 4.052 |
## Not.evening -0.169 0.768 0.055 -4.052 |
## lunch 0.254 0.386 0.011 1.822 |
## Not.lunch -0.044 0.066 0.011 -1.822 |
## dinner 0.744 1.581 0.042 3.531 |
## Not.dinner -0.056 0.119 0.042 -3.531 |
## always 0.095 0.126 0.005 1.186 |
## Not.always -0.050 0.066 0.005 -1.186 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## tea.time | 0.236 0.000 0.003 |
## evening | 0.046 0.085 0.055 |
## lunch | 0.060 0.028 0.011 |
## dinner | 0.082 0.000 0.042 |
## always | 0.061 0.034 0.005 |
## home | 0.013 0.002 0.027 |
## work | 0.071 0.020 0.027 |
## tearoom | 0.326 0.021 0.027 |
## friends | 0.201 0.058 0.024 |
## resto | 0.155 0.007 0.030 |# visualize MCA
plot(mca, invisible=c("ind"), graph.type = "classic",col.var = rainbow(ncol(tea_new_2)))
# It allows us to visualize many different categories from many catgeorical variables, all at once to see the categories that fit together and suggest association (similarities in behavior). For instance, unpackages and tea shop seem close to each other and separated from the rest of the categories. It suggests that they share some commonality / similar patterns or behavior in term of users preferences. Besides, they tend to occur together within the dataset.
# Select smaller dataset to do the MCA analyze with less variables
# let's now choose only few variables that could have correlations together but also show distinct behavior between users.
tea_new_2 %>%
pivot_longer(cols = everything()) %>%
ggplot(aes(x = value)) +
facet_wrap(~ name, scales = "free") +
geom_bar() +
labs(title = "Distribution of Categorical Variables") +
theme(axis.text.x = element_text(angle = 25, hjust = 1, size = 10)) +
theme(strip.text = element_text(size = 10))
# smaller data set - I will look at 2 different ones.
tea_smaller <- tea_new %>% dplyr::select(dinner, effect.on.health, frequency, evening, home, healthy, how, lunch, pub, price,Tea, where, work, sex)
tea_smaller2 <- tea_new %>% dplyr::select(diuretic, feminine, healthy, exciting, iron.absorption, escape.exoticism, relaxing, sophisticated, sex, slimming)
# MCA models
mca2 <- MCA(tea_smaller, graph = FALSE)
mca3 <- MCA(tea_smaller2, graph = FALSE)
# summary of the models
summary(mca2)##
## Call:
## MCA(X = tea_smaller, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6 Dim.7
## Variance 0.154 0.151 0.109 0.100 0.092 0.088 0.081
## % of var. 9.395 9.176 6.648 6.066 5.627 5.331 4.944
## Cumulative % of var. 9.395 18.572 25.220 31.286 36.914 42.245 47.189
## Dim.8 Dim.9 Dim.10 Dim.11 Dim.12 Dim.13 Dim.14
## Variance 0.081 0.075 0.071 0.068 0.064 0.063 0.061
## % of var. 4.911 4.595 4.306 4.129 3.887 3.820 3.690
## Cumulative % of var. 52.100 56.695 61.000 65.129 69.016 72.836 76.526
## Dim.15 Dim.16 Dim.17 Dim.18 Dim.19 Dim.20 Dim.21
## Variance 0.059 0.054 0.050 0.046 0.045 0.040 0.035
## % of var. 3.589 3.297 3.035 2.806 2.766 2.413 2.110
## Cumulative % of var. 80.114 83.411 86.446 89.252 92.018 94.431 96.542
## Dim.22 Dim.23
## Variance 0.033 0.024
## % of var. 2.020 1.438
## Cumulative % of var. 98.562 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.251 0.136 0.026 | 0.542 0.649 0.120 | -0.166
## 2 | -0.235 0.120 0.070 | 0.139 0.042 0.024 | -0.231
## 3 | -0.181 0.070 0.019 | -0.077 0.013 0.003 | -0.043
## 4 | -0.158 0.054 0.015 | 0.447 0.442 0.124 | -0.200
## 5 | -0.107 0.025 0.013 | 0.020 0.001 0.000 | 0.383
## 6 | -0.266 0.153 0.029 | 0.673 1.001 0.185 | -0.216
## 7 | -0.175 0.066 0.029 | 0.098 0.021 0.009 | 0.148
## 8 | -0.125 0.034 0.014 | 0.139 0.042 0.017 | 0.213
## 9 | 0.447 0.431 0.163 | -0.239 0.126 0.046 | 0.076
## 10 | 0.559 0.675 0.229 | -0.326 0.234 0.077 | -0.156
## ctr cos2
## 1 0.085 0.011 |
## 2 0.164 0.068 |
## 3 0.006 0.001 |
## 4 0.122 0.025 |
## 5 0.447 0.173 |
## 6 0.142 0.019 |
## 7 0.067 0.021 |
## 8 0.139 0.039 |
## 9 0.018 0.005 |
## 10 0.075 0.018 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr cos2 v.test
## dinner | 0.590 1.129 0.026 2.801 | 1.055 3.690 0.084 5.004
## Not.dinner | -0.044 0.085 0.026 -2.801 | -0.079 0.278 0.084 -5.004
## effect on health | 0.315 1.012 0.028 2.896 | 0.415 1.793 0.049 3.808
## No.effect on health | -0.089 0.286 0.028 -2.896 | -0.117 0.506 0.049 -3.808
## +2/day | 0.138 0.373 0.014 2.044 | -0.578 6.698 0.245 -8.562
## 1 to 2/week | 0.096 0.062 0.002 0.687 | 0.889 5.494 0.136 6.374
## 1/day | -0.311 1.416 0.045 -3.659 | 0.467 3.277 0.101 5.502
## 3 to 6/week | 0.229 0.275 0.007 1.416 | -0.298 0.477 0.011 -1.842
## evening | 0.132 0.276 0.009 1.648 | -0.277 1.246 0.040 -3.460
## Not.evening | -0.069 0.144 0.009 -1.648 | 0.145 0.651 0.040 3.460
## Dim.3 ctr cos2 v.test
## dinner | -0.266 0.323 0.005 -1.260 |
## Not.dinner | 0.020 0.024 0.005 1.260 |
## effect on health | 0.787 8.908 0.175 7.226 |
## No.effect on health | -0.222 2.513 0.175 -7.226 |
## +2/day | -0.225 1.405 0.037 -3.338 |
## 1 to 2/week | 0.948 8.623 0.155 6.797 |
## 1/day | -0.451 4.212 0.094 -5.309 |
## 3 to 6/week | 0.875 5.669 0.098 5.407 |
## evening | 0.433 4.202 0.098 5.409 |
## Not.evening | -0.226 2.197 0.098 -5.409 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## dinner | 0.026 0.084 0.005 |
## effect.on.health | 0.028 0.049 0.175 |
## frequency | 0.046 0.337 0.304 |
## evening | 0.009 0.040 0.098 |
## home | 0.001 0.023 0.035 |
## healthy | 0.015 0.016 0.489 |
## how | 0.587 0.297 0.002 |
## lunch | 0.000 0.152 0.049 |
## pub | 0.000 0.167 0.008 |
## price | 0.617 0.224 0.148 |summary(mca3)##
## Call:
## MCA(X = tea_smaller2, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6 Dim.7
## Variance 0.172 0.133 0.119 0.113 0.101 0.094 0.079
## % of var. 17.193 13.350 11.935 11.267 10.062 9.371 7.921
## Cumulative % of var. 17.193 30.543 42.478 53.744 63.806 73.177 81.098
## Dim.8 Dim.9 Dim.10
## Variance 0.070 0.064 0.055
## % of var. 7.014 6.369 5.519
## Cumulative % of var. 88.112 94.481 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.705 0.964 0.496 | -0.036 0.003 0.001 | 0.267
## 2 | -0.061 0.007 0.004 | 0.510 0.649 0.266 | 0.119
## 3 | -0.160 0.050 0.034 | -0.400 0.400 0.213 | 0.317
## 4 | -0.333 0.214 0.157 | -0.289 0.208 0.118 | 0.218
## 5 | -0.512 0.509 0.251 | -0.050 0.006 0.002 | 0.263
## 6 | -0.666 0.860 0.494 | -0.410 0.420 0.187 | 0.415
## 7 | -0.666 0.860 0.494 | -0.410 0.420 0.187 | 0.415
## 8 | 0.165 0.053 0.041 | -0.600 0.899 0.542 | -0.272
## 9 | -0.636 0.783 0.462 | -0.272 0.185 0.085 | -0.150
## 10 | -0.431 0.361 0.224 | -0.253 0.159 0.077 | 0.606
## ctr cos2
## 1 0.198 0.071 |
## 2 0.039 0.014 |
## 3 0.281 0.133 |
## 4 0.133 0.067 |
## 5 0.193 0.066 |
## 6 0.481 0.192 |
## 7 0.481 0.192 |
## 8 0.207 0.112 |
## 9 0.063 0.026 |
## 10 1.025 0.441 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr cos2
## diuretic | 0.409 5.643 0.231 8.311 | 0.241 2.533 0.081
## Not.diuretic | -0.565 7.792 0.231 -8.311 | -0.333 3.498 0.081
## feminine | 0.799 15.957 0.481 11.997 | -0.147 0.694 0.016
## Not.feminine | -0.603 12.038 0.481 -11.997 | 0.111 0.524 0.016
## healthy | 0.239 2.323 0.133 6.309 | -0.120 0.757 0.034
## Not.healthy | -0.557 5.421 0.133 -6.309 | 0.280 1.766 0.034
## exciting | 0.069 0.108 0.003 0.951 | 0.993 28.589 0.622
## No.exciting | -0.044 0.068 0.003 -0.951 | -0.626 18.023 0.622
## iron absorption | 0.767 3.536 0.068 4.503 | 0.130 0.130 0.002
## Not.iron absorption | -0.088 0.407 0.068 -4.503 | -0.015 0.015 0.002
## v.test Dim.3 ctr cos2 v.test
## diuretic 4.906 | 0.277 3.731 0.106 5.630 |
## Not.diuretic -4.906 | -0.383 5.153 0.106 -5.630 |
## feminine -2.205 | -0.373 5.022 0.105 -5.607 |
## Not.feminine 2.205 | 0.282 3.789 0.105 5.607 |
## healthy -3.173 | 0.369 7.977 0.317 9.741 |
## Not.healthy 3.173 | -0.861 18.614 0.317 -9.741 |
## exciting 13.640 | -0.133 0.575 0.011 -1.829 |
## No.exciting -13.640 | 0.084 0.363 0.011 1.829 |
## iron absorption 0.762 | -0.065 0.036 0.000 -0.379 |
## Not.iron absorption -0.762 | 0.007 0.004 0.000 0.379 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## diuretic | 0.231 0.081 0.106 |
## feminine | 0.481 0.016 0.105 |
## healthy | 0.133 0.034 0.317 |
## exciting | 0.003 0.622 0.011 |
## iron.absorption | 0.068 0.002 0.000 |
## escape.exoticism | 0.053 0.029 0.000 |
## relaxing | 0.006 0.438 0.062 |
## sophisticated | 0.173 0.002 0.106 |
## sex | 0.304 0.070 0.240 |
## slimming | 0.267 0.041 0.245 |# visualize MCA
plot(mca2, invisible=c("ind"), graph.type = "classic",col.var = rainbow(ncol(tea_new_2)))
plot(mca3, invisible=c("ind"), graph.type = "classic",col.var = rainbow(ncol(tea_new_2)))
Conclusion of MCAs
MCA 2: The categories “tea shop,” “upscale price,” and “unpackaged” appear closely related, indicating they collectively represent a particular type of behavior or consumer preference. Similarly, “tea bag” and “chain store” seem to indicate another pattern within the dataset.
MCA 3: Categories associated with tea benefits like “iron absorption,” “healthy,” “sophisticated,” and “diuretic” form a cluster, suggesting they share similar characteristics or appeal to a specific segment of consumers. Oppositely, tea without special benefits stands apart, indicating a different category or behavior. Moreover, the two categories “exciting” and “not relaxing” but also the two categories “feminine tea” and “female” show similarities, aligning with their expected conceptual correspondence in this context.
Note that the echo = FALSE parameter was added to the
code chunk to prevent printing of the R code that generated the
plot.
{r)
Chapter 5
Data wrangling
Load the data sets (BPRS and RATS) and analyze the dataset
library(tidyverse)
library(tidyr)
library(ggplot2)
# Save the url in a object
url_data1 <- "https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/BPRS.txt"
url_data2 <- "https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt"
# read the data and save it as an object
BPRS <- read.table(url_data1, header = TRUE, sep = ' ')
RATS <- read.table(url_data2, sep ='\t', header = T)Check the variable names, view the data contents and structures, and create some brief summaries of the variables , so that you understand the point of the wide form data. (1 point)
#BPRS dataset
str(BPRS)## 'data.frame': 40 obs. of 11 variables:
## $ treatment: int 1 1 1 1 1 1 1 1 1 1 ...
## $ subject : int 1 2 3 4 5 6 7 8 9 10 ...
## $ week0 : int 42 58 54 55 72 48 71 30 41 57 ...
## $ week1 : int 36 68 55 77 75 43 61 36 43 51 ...
## $ week2 : int 36 61 41 49 72 41 47 38 39 51 ...
## $ week3 : int 43 55 38 54 65 38 30 38 35 55 ...
## $ week4 : int 41 43 43 56 50 36 27 31 28 53 ...
## $ week5 : int 40 34 28 50 39 29 40 26 22 43 ...
## $ week6 : int 38 28 29 47 32 33 30 26 20 43 ...
## $ week7 : int 47 28 25 42 38 27 31 25 23 39 ...
## $ week8 : int 51 28 24 46 32 25 31 24 21 32 ...glimpse(BPRS)## Rows: 40
## Columns: 11
## $ treatment <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
## $ subject <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1…
## $ week0 <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, 38, 66, …
## $ week1 <int> 36, 68, 55, 77, 75, 43, 61, 36, 43, 51, 34, 52, 32, 35, 68, …
## $ week2 <int> 36, 61, 41, 49, 72, 41, 47, 38, 39, 51, 34, 49, 36, 36, 65, …
## $ week3 <int> 43, 55, 38, 54, 65, 38, 30, 38, 35, 55, 41, 54, 31, 34, 49, …
## $ week4 <int> 41, 43, 43, 56, 50, 36, 27, 31, 28, 53, 36, 48, 25, 25, 36, …
## $ week5 <int> 40, 34, 28, 50, 39, 29, 40, 26, 22, 43, 36, 43, 25, 27, 32, …
## $ week6 <int> 38, 28, 29, 47, 32, 33, 30, 26, 20, 43, 38, 37, 21, 25, 27, …
## $ week7 <int> 47, 28, 25, 42, 38, 27, 31, 25, 23, 39, 36, 36, 19, 26, 30, …
## $ week8 <int> 51, 28, 24, 46, 32, 25, 31, 24, 21, 32, 36, 31, 22, 26, 37, …dim(BPRS)## [1] 40 11summary(BPRS)## treatment subject week0 week1 week2
## Min. :1.0 Min. : 1.00 Min. :24.00 Min. :23.00 Min. :26.0
## 1st Qu.:1.0 1st Qu.: 5.75 1st Qu.:38.00 1st Qu.:35.00 1st Qu.:32.0
## Median :1.5 Median :10.50 Median :46.00 Median :41.00 Median :38.0
## Mean :1.5 Mean :10.50 Mean :48.00 Mean :46.33 Mean :41.7
## 3rd Qu.:2.0 3rd Qu.:15.25 3rd Qu.:58.25 3rd Qu.:54.25 3rd Qu.:49.0
## Max. :2.0 Max. :20.00 Max. :78.00 Max. :95.00 Max. :75.0
## week3 week4 week5 week6 week7
## Min. :24.00 Min. :20.00 Min. :20.00 Min. :19.00 Min. :18.0
## 1st Qu.:29.75 1st Qu.:28.00 1st Qu.:26.00 1st Qu.:22.75 1st Qu.:23.0
## Median :36.50 Median :34.50 Median :30.50 Median :28.50 Median :30.0
## Mean :39.15 Mean :36.35 Mean :32.55 Mean :31.23 Mean :32.2
## 3rd Qu.:44.50 3rd Qu.:43.00 3rd Qu.:38.00 3rd Qu.:37.00 3rd Qu.:38.0
## Max. :76.00 Max. :66.00 Max. :64.00 Max. :64.00 Max. :62.0
## week8
## Min. :20.00
## 1st Qu.:22.75
## Median :28.00
## Mean :31.43
## 3rd Qu.:35.25
## Max. :75.00The data shows analyze of BPRS for different participants (20). There are no groups for participants, but we have 2 types of treatment which we can consider as 2 groups. The max BPRS over the weeks is 95 and the min is 18. The PBRS changes every day in term of min, max, median and mean. The variables treatment and subject are integer -> need to be changed to factor for further analyzes. We have 40 rows 40 and 11 columns -> format is wide -> needs to be changed to long format.
#RATS
str(RATS)## 'data.frame': 16 obs. of 13 variables:
## $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
## $ Group: int 1 1 1 1 1 1 1 1 2 2 ...
## $ WD1 : int 240 225 245 260 255 260 275 245 410 405 ...
## $ WD8 : int 250 230 250 255 260 265 275 255 415 420 ...
## $ WD15 : int 255 230 250 255 255 270 260 260 425 430 ...
## $ WD22 : int 260 232 255 265 270 275 270 268 428 440 ...
## $ WD29 : int 262 240 262 265 270 275 273 270 438 448 ...
## $ WD36 : int 258 240 265 268 273 277 274 265 443 460 ...
## $ WD43 : int 266 243 267 270 274 278 276 265 442 458 ...
## $ WD44 : int 266 244 267 272 273 278 271 267 446 464 ...
## $ WD50 : int 265 238 264 274 276 284 282 273 456 475 ...
## $ WD57 : int 272 247 268 273 278 279 281 274 468 484 ...
## $ WD64 : int 278 245 269 275 280 281 284 278 478 496 ...glimpse(RATS)## Rows: 16
## Columns: 13
## $ ID <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ Group <int> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ WD1 <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 555, 470,…
## $ WD8 <int> 250, 230, 250, 255, 260, 265, 275, 255, 415, 420, 445, 560, 465,…
## $ WD15 <int> 255, 230, 250, 255, 255, 270, 260, 260, 425, 430, 450, 565, 475,…
## $ WD22 <int> 260, 232, 255, 265, 270, 275, 270, 268, 428, 440, 452, 580, 485,…
## $ WD29 <int> 262, 240, 262, 265, 270, 275, 273, 270, 438, 448, 455, 590, 487,…
## $ WD36 <int> 258, 240, 265, 268, 273, 277, 274, 265, 443, 460, 455, 597, 493,…
## $ WD43 <int> 266, 243, 267, 270, 274, 278, 276, 265, 442, 458, 451, 595, 493,…
## $ WD44 <int> 266, 244, 267, 272, 273, 278, 271, 267, 446, 464, 450, 595, 504,…
## $ WD50 <int> 265, 238, 264, 274, 276, 284, 282, 273, 456, 475, 462, 612, 507,…
## $ WD57 <int> 272, 247, 268, 273, 278, 279, 281, 274, 468, 484, 466, 618, 518,…
## $ WD64 <int> 278, 245, 269, 275, 280, 281, 284, 278, 478, 496, 472, 628, 525,…dim(RATS)## [1] 16 13summary(RATS)## ID Group WD1 WD8 WD15
## Min. : 1.00 Min. :1.00 Min. :225.0 Min. :230.0 Min. :230.0
## 1st Qu.: 4.75 1st Qu.:1.00 1st Qu.:252.5 1st Qu.:255.0 1st Qu.:255.0
## Median : 8.50 Median :1.50 Median :340.0 Median :345.0 Median :347.5
## Mean : 8.50 Mean :1.75 Mean :365.9 Mean :369.1 Mean :372.5
## 3rd Qu.:12.25 3rd Qu.:2.25 3rd Qu.:480.0 3rd Qu.:476.2 3rd Qu.:486.2
## Max. :16.00 Max. :3.00 Max. :555.0 Max. :560.0 Max. :565.0
## WD22 WD29 WD36 WD43
## Min. :232.0 Min. :240.0 Min. :240.0 Min. :243.0
## 1st Qu.:267.2 1st Qu.:268.8 1st Qu.:267.2 1st Qu.:269.2
## Median :351.5 Median :356.5 Median :360.0 Median :360.0
## Mean :379.2 Mean :383.9 Mean :387.0 Mean :386.0
## 3rd Qu.:492.5 3rd Qu.:497.8 3rd Qu.:504.2 3rd Qu.:501.0
## Max. :580.0 Max. :590.0 Max. :597.0 Max. :595.0
## WD44 WD50 WD57 WD64
## Min. :244.0 Min. :238.0 Min. :247.0 Min. :245.0
## 1st Qu.:270.0 1st Qu.:273.8 1st Qu.:273.8 1st Qu.:278.0
## Median :362.0 Median :370.0 Median :373.5 Median :378.0
## Mean :388.3 Mean :394.6 Mean :398.6 Mean :404.1
## 3rd Qu.:510.5 3rd Qu.:516.0 3rd Qu.:524.5 3rd Qu.:530.8
## Max. :595.0 Max. :612.0 Max. :618.0 Max. :628.0The data shows weight for different rats (16) at different time from 1 to 64. Each rat belongs to a different group. There are 3 groups. The max weight over the weeks is 628 and the min weight is 225. The mean changes every day in term of min, max, median and mean. The variables ID and group are integer -> need to be changed to factor for further analyzes. There are 16 rows and 13 columns -> format is wide and needs to be changed to long format.
2. Convert the categorical variables to factors
#BPRS
BPRS$treatment <- factor(BPRS$treatment)
BPRS$subject <- factor(BPRS$subject)
# RATS
RATS$ID <- factor(RATS$ID )
RATS$Group <- factor(RATS$Group)3. Convert the table to long form
Add a week variable to BPRS and a Time variable to RATS > reorder by period of time (week or WD) > add one column where the week or WD is an integer.
BPRS
BPRS_long <- pivot_longer(BPRS, cols = -c(treatment, subject), # columns are not pivoted, they are returned as they are
names_to = "weeks", values_to = "bprs") %>% # names_to is the new column created for the former column titles, the values_to is the new column created for the values
arrange(weeks) %>% # order by weeks variable
mutate(week = as.integer(substr(weeks,5,5)))RATS
RATS_long <- pivot_longer(RATS, cols = -c(ID, Group),
names_to = "WD", values_to = "weight") %>%
arrange(WD) %>%
mutate(Time = as.integer(substr(WD,3,4)))Compare wide vs long format
Now, take a serious look at the new data sets and compare them with their wide form versions: Check the variable names, view the data contents and structures, and create some brief summaries of the variables. Make sure that you understand the point of the long form data and the crucial difference between the wide and the long forms before proceeding the to Analysis exercise. (2 points)
Long format data allow for a more detailed examination of individual measurements, making it easier to assess the impact of various factors on the outcome. This format also facilitates the application of statistical methods designed for longitudinal or repeated measures analysis.
BPRS
str(BPRS_long)## tibble [360 × 5] (S3: tbl_df/tbl/data.frame)
## $ treatment: Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
## $ subject : Factor w/ 20 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ weeks : chr [1:360] "week0" "week0" "week0" "week0" ...
## $ bprs : int [1:360] 42 58 54 55 72 48 71 30 41 57 ...
## $ week : int [1:360] 0 0 0 0 0 0 0 0 0 0 ...glimpse(BPRS_long)## Rows: 360
## Columns: 5
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1…
## $ weeks <chr> "week0", "week0", "week0", "week0", "week0", "week0", "week0…
## $ bprs <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, 38, 66, …
## $ week <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …dim(BPRS_long)## [1] 360 5summary(BPRS_long)## treatment subject weeks bprs week
## 1:180 1 : 18 Length:360 Min. :18.00 Min. :0
## 2:180 2 : 18 Class :character 1st Qu.:27.00 1st Qu.:2
## 3 : 18 Mode :character Median :35.00 Median :4
## 4 : 18 Mean :37.66 Mean :4
## 5 : 18 3rd Qu.:43.00 3rd Qu.:6
## 6 : 18 Max. :95.00 Max. :8
## (Other):252The main difference with the long table is that we now have one entry for each measure for a week, subject, treatment type like if each entry would be an independent one. We can now consider the impact of variables like treatment types, weeks, subjects and their BPRS score. We now have one mean, and one median for the total of the study. Treatment and subject are now factor variables (dummy variables). This transformation helps represent categorical information in a format that can be used effectively in statistical analyses. Week is now an integer. This allows for mathematical operations and analysis involving these variables.
RATS
str(RATS_long)## tibble [176 × 5] (S3: tbl_df/tbl/data.frame)
## $ ID : Factor w/ 16 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ Group : Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 1 2 2 ...
## $ WD : chr [1:176] "WD1" "WD1" "WD1" "WD1" ...
## $ weight: int [1:176] 240 225 245 260 255 260 275 245 410 405 ...
## $ Time : int [1:176] 1 1 1 1 1 1 1 1 1 1 ...glimpse(RATS_long)## Rows: 176
## Columns: 5
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3,…
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1, 1, …
## $ WD <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", …
## $ weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 555, 470…
## $ Time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 15, 15, 15,…dim(RATS_long)## [1] 176 5summary(RATS_long)## ID Group WD weight Time
## 1 : 11 1:88 Length:176 Min. :225.0 Min. : 1.00
## 2 : 11 2:44 Class :character 1st Qu.:267.0 1st Qu.:15.00
## 3 : 11 3:44 Mode :character Median :344.5 Median :36.00
## 4 : 11 Mean :384.5 Mean :33.55
## 5 : 11 3rd Qu.:511.2 3rd Qu.:50.00
## 6 : 11 Max. :628.0 Max. :64.00
## (Other):110The main difference with the long table is that we now have one entry for each measure for a time, rat, group like if each entry would be an independent one. We can now do analyzes like if we would have only one rat and different situations based on variables like group, time which impact the rat’s weight. We now have one mean, and one median for the total of the study: 384.5 mean, 344.5 median. ID and Group are now factor variables (dummy variables). This transformation helps represent categorical information in a format that can be used effectively in statistical analyses. Time is now an integer. This allows for mathematical operations and analysis involving these variables.
Model with a baseline as a covariate - dataset RATS
Implement the analyses of Chapter 8 of MABS, using the R codes of Exercise Set 6: Meet and Repeat: PART I but using the RATS data (from Chapter 9 and Meet and Repeat: PART II). (0-7 points: 0-4 points for graphs or analysis results + 0-3 points for their interpretations)
Plot the data
#Access the packages
library(ggplot2)
library(tidyr)
# Draw the plot
ggplot(RATS_long, aes(x = Time, y = weight, linetype = ID)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ Group, labeller = label_both) +
theme(legend.position = "none") 
We can see a huge difference between each group in term of weight from the beginning and this difference seems to stay over time so data points might be dependent to each other. The group 2 seems to have one rat’s weight very different than the others. Because of the massive difference let’s standardize the data so that the starting point does not influence the results and compare groups with standardized data.
Analyze of standardized means of rat’s weight per group and per Time
# Standardise the variable weight
RATS2 <- RATS_long %>%
group_by(Time) %>% # perform the next code lines per week
mutate(stdweight = (weight - mean(weight)) / sd(weight)) %>% # add up the bprs result per week
ungroup() # ungroup for the next use of this object.
# Glimpse the data
glimpse(RATS2)## Rows: 176
## Columns: 6
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2,…
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1, …
## $ WD <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1…
## $ weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 555, …
## $ Time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 15, 15, …
## $ stdweight <dbl> -1.0011429, -1.1203857, -0.9613953, -0.8421525, -0.8819001, …# Draw the plot
ggplot(RATS2, aes(x = Time, y = stdweight, linetype = ID)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ Group, labeller = label_both) +
theme(legend.position = "none") 
Analyze of mean and standard deviation of the rats weights per group and per Time.
# Summary data with mean and standard error of weight
RATS3 <- RATS_long %>%
group_by(Group, Time) %>%
summarise(mean = mean(weight),
sd = sd(weight),
n = n()) %>%
mutate(se = sd / sqrt(n)) %>%
ungroup()## `summarise()` has grouped output by 'Group'. You can override using the
## `.groups` argument.# Glimpse the data
glimpse(RATS3)## Rows: 33
## Columns: 6
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2…
## $ Time <int> 1, 8, 15, 22, 29, 36, 43, 44, 50, 57, 64, 1, 8, 15, 22, 29, 36, …
## $ mean <dbl> 250.625, 255.000, 254.375, 261.875, 264.625, 265.000, 267.375, 2…
## $ sd <dbl> 15.22158, 13.09307, 11.47591, 13.60081, 11.05748, 11.78377, 10.9…
## $ n <int> 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4…
## $ se <dbl> 5.381640, 4.629100, 4.057346, 4.808614, 3.909409, 4.166190, 3.87…# Plot the mean per group of rats
ggplot(RATS3, aes(x = Time, y = mean, linetype = Group, shape = Group, color = Group)) +
geom_line() +
scale_linetype_manual(values = c(1,2,3)) + # set the 3 different lines for the 3 groups
geom_point(size=3) +
scale_shape_manual(values = c(1,2,3)) + # set the 3 different point shapes for the 3 groups
geom_errorbar(aes(ymin=mean-se, ymax=mean+se, linetype="1"), width=0.3) +
theme(legend.position = c(0.8,0.8)) +
scale_y_continuous(name = "mean(bprs) +/- se(bprs)")
This plot shos well the difference of weight per group but also withing group. It also shows well that the evolution over time depends on the starting point weight of a rat.
Analyze of weights mean and standard deviation per group
# Create a summary data by group and ID with mean as the summary variable (ignoring baseline Time 0)
RATS4 <- RATS_long %>%
filter(Time > 1) %>% # post first day.
group_by(Group, ID) %>%
summarise( mean=mean(weight) ) %>%
ungroup()## `summarise()` has grouped output by 'Group'. You can override using the
## `.groups` argument.# Glimpse the data
glimpse(RATS4)## Rows: 16
## Columns: 3
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ mean <dbl> 263.2, 238.9, 261.7, 267.2, 270.9, 276.2, 274.6, 267.5, 443.9, 4…# Draw a boxplot of the mean weight vs group
ggplot(RATS4, aes(x = Group, y = mean)) +
geom_boxplot() +
stat_summary(fun = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "mean(bprs) for the studied Time period")
We can see outliers in group 1 - a very small rat We can see an outlier in group 2 - a very big rat. We can see an outlier in group 3 - a smaller rat than the rest We can also notify the big differences between groups and between rats. It is possible to filter the outliers but this needs to be carefully done, especially if they represent natural variation within the data. Outliers can be legitimate data points that may carry important information or represent unique cases (biological variability for instance). However, if they are due to measurement errors or anomalies, it might be appropriate to handle them. In that case, let’s consider that the outliers in group 1 and 3 are a result of a rare and unrepeatable event that substantially skews the observed results but doesn’t reflect the studied phenomenon’s reality.
Analyze of weights mean and standard deviation per group- Removing the outliers
# Create a new data by filtering the outliers and adjust the ggplot code the draw the plot again with the new data
RATS4_2 <- RATS4 %>%
filter(mean > 240) %>%
filter(mean < 540)
ggplot(RATS4_2, aes(x = Group, y = mean)) +
geom_boxplot() +
stat_summary(fun = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "for the studied Time period")
After outliers removal, the plot exhibits a more normally distributed pattern, making it easier to visualize trends and patterns in the data.
Let’s create a linear model
# Save the Time 0 weigh of rats as an object to use it in the model
RATS4_3 <- RATS4 %>%
mutate(baseline = RATS$WD1)
# Fit the linear model with the mean as the response (variables to influence weight are the baseline: situation in WD1 and the Group)
# We don't consider the rats individuality but more the group differences that impact the results but also their weight at the beginning of the study
fit_RATS <- lm(mean ~ baseline + Group, data = RATS4_3)
# summary of the model
summary(fit_RATS)##
## Call:
## lm(formula = mean ~ baseline + Group, data = RATS4_3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -23.905 -4.194 2.190 7.577 14.800
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.16375 21.87657 1.516 0.1554
## baseline 0.92513 0.08572 10.793 1.56e-07 ***
## Group2 34.85753 18.82308 1.852 0.0888 .
## Group3 23.67526 23.25324 1.018 0.3287
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 11.68 on 12 degrees of freedom
## Multiple R-squared: 0.9936, Adjusted R-squared: 0.992
## F-statistic: 622.1 on 3 and 12 DF, p-value: 1.989e-13anova(fit_RATS)## Analysis of Variance Table
##
## Response: mean
## Df Sum Sq Mean Sq F value Pr(>F)
## baseline 1 253625 253625 1859.8201 1.57e-14 ***
## Group 2 879 439 3.2219 0.07586 .
## Residuals 12 1636 136
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1AIC(fit_RATS) # 129## [1] 129.4492BIC(fit_RATS) # 133## [1] 133.3121plot(fit_RATS)
# just for the try, let's remove baseline and see what is happening
# this model explain 92% but it is because there is a strong dependence between the data from one group and from each rat. Therefore there is a correlation that is not justified. This correlation might indicate interdependence that are not justified.
fit_RATS2 <- lm(mean ~ Group, data = RATS4_3)
summary(fit_RATS2)##
## Call:
## lm(formula = mean ~ Group, data = RATS4_3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -43.900 -27.169 2.325 9.069 106.200
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 265.02 12.98 20.419 2.92e-11 ***
## Group2 222.78 22.48 9.909 2.00e-07 ***
## Group3 262.48 22.48 11.675 2.90e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 36.71 on 13 degrees of freedom
## Multiple R-squared: 0.9316, Adjusted R-squared: 0.9211
## F-statistic: 88.52 on 2 and 13 DF, p-value: 2.679e-08anova(fit_RATS2)## Analysis of Variance Table
##
## Response: mean
## Df Sum Sq Mean Sq F value Pr(>F)
## Group 2 238620 119310 88.525 2.679e-08 ***
## Residuals 13 17521 1348
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1AIC(fit_RATS2) # 165## [1] 165.383BIC(fit_RATS2) # 168## [1] 168.4733FIRST MODEL WITH BASELINE
- The T value of the baseline is 10.8, it is very high so it means this baseline impacts the studies weight (p is very small so it is significant). The estimate for the baseline variable remains significantly different from zero, even when considering its standard error, reinforcing its importance in explaining weight changes.
- Group Impact: Group 1 serves as reference. The coefficients for Group2 and Group3 which represent the differences between those groups and the baseline, aren’t as substantial, and their impact on weight might not be statistically significant based on the p-values.
- The spread of residuals looks decent, with the majority being relatively close to zero. However, there are a few larger residuals (especially the maximum value - the median is 2.19, and the values between the 1Q and 3Q range from -4.194 to 7.577), indicating potential areas where the model might not fit perfectly and where the fitted values are largely underestimated (but also overestimated). This is also very visible when analyzing the residuals plot.
- Similarly, the anova results suggests that the baseline has a strong and significant effect on the mean. Group might have some influence and shows a p-value of 0.076, indicating a relatively high significance threshold.
Random Effects in Longitudinal Analysis - BPRS dataset
Implement the analyses of Chapter 9 of MABS, using the R codes of Exercise Set 6: Meet and Repeat: PART II, but using the BPRS data (from Chapter 8 and Meet and Repeat: PART I). (0-8 points: 0-4 points for graphs or analysis results + 0-4 points for their interpretations)
Plot the data
# Draw a plot for each subject and its bprs per week
ggplot(BPRS_long, aes(x = week, y = bprs, linetype = subject)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ treatment, labeller = label_both) +
scale_y_continuous(name = "bprs over time")
Insights: Variations between subject are broad. We can see that most of the subject has a decreasing bprs over week no matter the treatment. Mostly the time and potentially the subject starting point seems to affect the result more than the treatment type itself. Treatment 2 subjects seem to have higher ‘pbrs’ from the beginning and it seems to influence the rest of the weeks.
Creation of a linear model
# create a simple regression model
BPRS_model1 <- lm(bprs ~ week + treatment, data = BPRS_long)
# print out a summary of the model
summary(BPRS_model1) ##
## Call:
## lm(formula = bprs ~ week + treatment, data = BPRS_long)
##
## Residuals:
## Min 1Q Median 3Q Max
## -22.454 -8.965 -3.196 7.002 50.244
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.4539 1.3670 33.982 <2e-16 ***
## week -2.2704 0.2524 -8.995 <2e-16 ***
## treatment2 0.5722 1.3034 0.439 0.661
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.37 on 357 degrees of freedom
## Multiple R-squared: 0.1851, Adjusted R-squared: 0.1806
## F-statistic: 40.55 on 2 and 357 DF, p-value: < 2.2e-16anova(BPRS_model1) ## Analysis of Variance Table
##
## Response: bprs
## Df Sum Sq Mean Sq F value Pr(>F)
## week 1 12372 12371.5 80.9143 <2e-16 ***
## treatment 1 29 29.5 0.1927 0.6609
## Residuals 357 54584 152.9
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1BIC(BPRS_model1) # 2852## [1] 2852.881AIC(BPRS_model1) # 2837## [1] 2837.337Insights: the model explains only 18% of the variance which is very small. It suggests that a significant portion of the variability in the ‘bprs’ remains unexplained by the predictors in the model. Week has a big effect on the BPRS and It is significant. The high residual variability indicates the model is not effective to explain the ‘bprs’. It may be worth exploring other factors that could better explain the variability in ‘bprs’.
Adding random effects to the model:
Notes:
- Fixed Effects: Variables we are interested in and we want to estimate the effects or coefficients directly. They have a systematic impact on the response variable. Fixed effects are often the primary focus of analysis, while random effects account for additional variability that isn’t explicitly modeled as a primary predictor but impacts the variance.
- Random Effects: Variables we think impact the variability but not in a systemic manner.
For example, in a study analyzing subject’s ‘bprs’ based on different treatment, the subjects themselves might be included as a random effect because we’re not directly interested in the effect of the subjects, but we acknowledge that they may introduce variability in their ‘bprs’ that is not systematically related to other predictors.
library(lme4) # to add random effects## Loading required package: Matrix##
## Attaching package: 'Matrix'## The following objects are masked from 'package:tidyr':
##
## expand, pack, unpacklibrary(lmerTest) # to get p value with random effects##
## Attaching package: 'lmerTest'## The following object is masked from 'package:lme4':
##
## lmer## The following object is masked from 'package:stats':
##
## step# Create a random intercept model with subject only. The week does not effect on the subject.
BPRS_model2 <- lmer(bprs ~ week + treatment + (1 | subject), data = BPRS_long, REML = FALSE)
# Summary of the model
summary(BPRS_model2)## Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
## method [lmerModLmerTest]
## Formula: bprs ~ week + treatment + (1 | subject)
## Data: BPRS_long
##
## AIC BIC logLik deviance df.resid
## 2748.7 2768.1 -1369.4 2738.7 355
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0481 -0.6749 -0.1361 0.4813 3.4855
##
## Random effects:
## Groups Name Variance Std.Dev.
## subject (Intercept) 47.41 6.885
## Residual 104.21 10.208
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 46.4539 1.9090 37.2392 24.334 <2e-16 ***
## week -2.2704 0.2084 340.0000 -10.896 <2e-16 ***
## treatment2 0.5722 1.0761 340.0000 0.532 0.595
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.437
## treatment2 -0.282 0.000anova(BPRS_model2) ## Type III Analysis of Variance Table with Satterthwaite's method
## Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
## week 12371.5 12371.5 1 340 118.7136 <2e-16 ***
## treatment 29.5 29.5 1 340 0.2828 0.5952
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1BIC(BPRS_model2) # 2768## [1] 2768.143AIC(BPRS_model2) # 2748## [1] 2748.712From the F values, it seems that the “week” factor has a much higher F value compared to the “treatment” factor, suggesting once again that the “week” factor explains more variability in the data compared to the “treatment” factor.
This model is bit better than the previous one when considering the BIC and AIC. This model does not take into consideration the fact that each subject may have different evolution of ‘bprs’ over time, therefore let’s try a new model that would take into consideration this random effect.
Mixed model with random effects - week and subject
- Individual Variability: If each subject indeed has different baseline levels (‘bprs’ at the start), a random intercept for ‘subject’ ((1 | subject)) might be appropriate.
- Varying Response to Time: If time affects individuals differently, incorporating random slopes for ‘week’ within ‘subject’ ((week | subject)) might be beneficial.
# create a random intercept and random slope model
library(lme4)
BPRS_model3 <- lmer(bprs ~ week + treatment + (week | subject), data = BPRS_long, REML = FALSE)
# summary of the model
summary(BPRS_model3)## Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
## method [lmerModLmerTest]
## Formula: bprs ~ week + treatment + (week | subject)
## Data: BPRS_long
##
## AIC BIC logLik deviance df.resid
## 2745.4 2772.6 -1365.7 2731.4 353
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.8919 -0.6194 -0.0691 0.5531 3.7976
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.8222 8.0512
## week 0.9609 0.9802 -0.51
## Residual 97.4305 9.8707
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 46.4539 2.1052 22.6795 22.066 < 2e-16 ***
## week -2.2704 0.2977 19.9991 -7.626 2.42e-07 ***
## treatment2 0.5722 1.0405 320.0005 0.550 0.583
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.582
## treatment2 -0.247 0.000# ANOVA test on the two models.
anova(BPRS_model2, BPRS_model3)## Data: BPRS_long
## Models:
## BPRS_model2: bprs ~ week + treatment + (1 | subject)
## BPRS_model3: bprs ~ week + treatment + (week | subject)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## BPRS_model2 5 2748.7 2768.1 -1369.4 2738.7
## BPRS_model3 7 2745.4 2772.6 -1365.7 2731.4 7.2721 2 0.02636 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# plot the residuals
plot(BPRS_model3)
In BPRS_model2: (1 | subject) represents a random intercept for each ‘subject’. The variance associated with this random intercept (subject-specific variability that is not explained by fixed effects) is estimated in this model.
In BPRS_model3: (week | subject) represents both random intercepts and random slopes for ‘week’ within each ‘subject’. This allows the ‘week’ effect to vary for each ‘subject’. This models seems to be better since we can see on the first plot that each individual has a very different ‘bprs’ slope per subject and this difference per subject exist over time.
Comparison between the 2 models:
- BPRS_model2 has 5 estimated parameters and a higher AIC (2748.7) and a lower BIC (2768.1).The BIC is lower because the BIC penalizes the more complex models (favorizing the model with 5 vs 7 parameters)
- BPRS_model3 has 7 estimated parameters and slightly lower AIC (2745.4) and a higher BIC (2772.6) compared to BPRS_model2.
- The chi-square test for model comparison results in a p-value of 0.02636 for the model3, indicating that BPRS_model3 significantly improves the model fit compared to BPRS_model2.
- BPRS_model3 has a higher log-likelihood (-1365.7) compared to BPRS_model2 (-1369.4). Therefore, based solely on the logLik values, BPRS_model3 appears to provide a better fit to the data than BPRS_model2.
- The lower deviance values of the BPRS_model3 implies that the model explains a larger proportion of the total variance in the data vs the other model.
To conclude, the BPRS_model3 seems to provide a better fit for the data compared to BPRS_model2 due to its lower AIC, deviance, higher logLik, and its significant improvement in model fit indicated by the chi-square test.
Model3 analyzes:
The model captures a significant effect of ‘week’ on ‘bprs’ scores, indicating a decrease in ‘bprs’ over time.
- The scaled residuals range from -2.8919 to 3.7976, indicating a relatively wide spread of values = heteroscedasticity in the residuals which we can also observe in the residuals plot. The scaled residuals are close to zero (around -0.0691) implies that a substantial portion of the observations have residuals near the mean. The negative scaled residuals (-2.8919) are less extreme compared to the positive scaled residuals (3.7976). It suggests that the model more often underestimates the ‘bprs’ (a positive residual (observed - predicted > 0) indicates an underestimation by the model.)
- Random Intercept (Subject-specific intercept): The estimated
variance of the intercept among subjects is 64.8222, which translates to
a standard deviation of about 8 points around the average BPRS score
(intercept is 45.4539 in fixed effects) when all other predictors does
not exist (week, treatment). This means that subjects may start with
‘bprs’ scores that differ by about 8 points on average from the overall
mean and it represents around 17.6% variability relative to the
intercept’s value in the BPRS scores. If we compare it to the mean, the
relative standard deviation is approximately 21.23%.
This understanding supports the notion that subjects’ characteristics
(not accounted for by other predictors) might influence their initial
‘bprs’ scores, and this influence could persist over time.
- The estimated variance of the effect of ‘week’ among subjects is 0.9609, with a standard deviation of 0.9802. The deviation suggests that for certain subjects, time does not significantly affect their ‘bprs’ scores but for others it does.
- Besides, there’s a negative correlation (-0.51) between the random intercept and the random slope for ‘week’ and ‘subjects’ together. The correlation indicates how much the effect on ‘bprs’ scores can vary between subjects and over time. The negative correlation suggests that subjects with higher initial ‘bprs’ tend to have a weaker decrease in ‘bprs’ over time (‘week’) compared to subjects with lower initial scores.
- Correlation of fixed effect between Intercept and ‘week’ (-0.58) is negative. As the starting/intercept ‘bprs’ scores increase, the effect of ‘week’ on ‘bprs’ scores appears to decrease. This implies that subjects with higher initial ‘bprs’ scores might experience a slower rate of change (‘week’ effect) compared to those with lower initial scores. -> similar conclusion with the random effects.
- Correlation between Intercept and ‘treatment 2’ (-0.247) is negative but a bit less than the first correlation. This indicates a moderate negative association between the effect of ‘week’ and the effect of ‘treatment2’ relative to the reference level ‘treatment1’. The presence of ‘treatment2’ might attenuate the impact of ‘week’ on ‘bprs’ scores compared to ‘treatment1’. Because we want to reduce the ‘bprs’, the ‘treatment2’ might limit the positive effect of time (‘week’) on improving ‘bprs’, therefore possibly leading to a less favorable outcome compared to ‘treatment1’. This effect might be even worst for those starting with a higher ‘pbrs’ as we saw that time affects them less in reducing the ‘bprs’. These conclusions would require further research to be validated.
Mixed model with random effects - week and subject + consider bprs change over time, based on treatment
Note: week and treatment both influences each other. We use the ‘*’ if there’s reason to believe that the effect of week on bprs differs if treatment is taken or not. If there’s no reason to believe that the relationship between week and bprs changes based on treatment, using the model week + treatment is sufficient.
# create a random intercept and random slope model with the interaction
BPRS_model4 <- lmer(bprs ~ week * treatment + (week | subject), data = BPRS_long, REML = FALSE)
# print a summary of the model
summary(BPRS_model4)## Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
## method [lmerModLmerTest]
## Formula: bprs ~ week * treatment + (week | subject)
## Data: BPRS_long
##
## AIC BIC logLik deviance df.resid
## 2744.3 2775.4 -1364.1 2728.3 352
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0512 -0.6271 -0.0768 0.5288 3.9260
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.9964 8.0620
## week 0.9687 0.9842 -0.51
## Residual 96.4707 9.8220
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 47.8856 2.2521 29.6312 21.262 < 2e-16 ***
## week -2.6283 0.3589 41.7201 -7.323 5.24e-09 ***
## treatment2 -2.2911 1.9090 319.9977 -1.200 0.2310
## week:treatment2 0.7158 0.4010 319.9977 1.785 0.0752 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) week trtmn2
## week -0.650
## treatment2 -0.424 0.469
## wek:trtmnt2 0.356 -0.559 -0.840plot(BPRS_model4)
# perform an ANOVA test on the 3 models
anova(BPRS_model2,BPRS_model3,BPRS_model4) ## Data: BPRS_long
## Models:
## BPRS_model2: bprs ~ week + treatment + (1 | subject)
## BPRS_model3: bprs ~ week + treatment + (week | subject)
## BPRS_model4: bprs ~ week * treatment + (week | subject)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## BPRS_model2 5 2748.7 2768.1 -1369.4 2738.7
## BPRS_model3 7 2745.4 2772.6 -1365.7 2731.4 7.2721 2 0.02636 *
## BPRS_model4 8 2744.3 2775.4 -1364.1 2728.3 3.1712 1 0.07495 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Chi square of new model is too high to be significant and to draw conclusion
Analyze the fitted vs observed values with the model 3
# Create a vector of the fitted values of model 3 - our best model
Fitted_bprs <- fitted(BPRS_model3)
# Create a new column fitted to our BPRS_long dataset
BPRS_new <- BPRS_long %>%
mutate(Fitted_bprs = Fitted_bprs)
# draw the plot for the subjects, observed vs fitted values:
ggplot(BPRS_long, aes(x = week)) +
geom_line(aes(y = bprs, group = subject, linetype = "Observed")) +
geom_line(aes(y = Fitted_bprs, group = subject, linetype = "Fitted")) +
facet_grid(. ~ treatment, labeller = label_both) +
labs(x = "Week", y = "bprs") 
Note: It does not seem to be such a good model to predict the BPRS of subject in each treatment group. The example we saw in the book with the RATS dataset was better.